Mathematics Interesting Mumbers





INTERESTING  NUMBERS  The interesting number paradox is a paradox that arises from attempting to classify natural numbers as "interesting" or "dull". The paradox states that all natural numbers are interesting. The "proof" is by contradiction.   If there were uninteresting numbers, there would be a smallest uninteresting number But the smallest uninteresting number is itself interesting by being so, producing a contradiction.

Leopold Kronecker said: " God himself made the integers: everything else is the work of man".

1729 is also known as the Hardy-Ramanujan number. Godfrey Hardy, an English mathematician was visiting Srinivasa Ramanujan in a hospital when Ramanujan told him about this number.

Here are Hardy's words: I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."   Notice that 1729 = 1^3 + 12^3 = 9^3 + 10^3.

Twin Primes  It is interesting that it cannot be proven that there are or there are not an infinite number of "twin primes" 

Similarly the Goldbach Conjecture states: Every even integer greater than 2 can be expressed as the sum of two primes. The conjecture has been shown to hold for all integers less than 4 × 1018 but remains unproven despite considerable effort.

The even integers from 4 to 28 as sums of two primes: Even integers correspond to horizontal lines. For each prime, there are two oblique lines, one red and one blue. The sums of two primes are the intersections of one red and one blue line, marked by a circle. Thus the circles on a given horizontal line give all partitions of the corresponding even integer into the sum of two primes.

111,111,111 x 111,111,111 = 12345678987654321

Some of the following comes from David Wells book: "The Penguin Dictionary of Curious and Interesting Numbers".  Also examine Patrick de Geest's website:

See more on numbers at

-1  This is e raised to i times pi. 

The Euler formula, sometimes also called the Euler identity  and can be derived  from   e^(ix)=cosx+isinx,    The special case of the formula with x=pi gives the beautiful identity


This equation connects the fundamental numbers ipie, 1, and 0 (zero), the fundamental operations +×, and exponentiation, the most important relation =, and nothing else. Gauss is reported to have commented that if this formula was not immediately obvious, the reader would never be a first-class mathematician.

The Euler formula can also be demonstrated using a series expansion

e^(ix) = sum_(n=0)^(infty)((ix)^n)/(n!)
= sum_(n=0)^(infty)((-1)^nx^(2n))/((2n)!)+isum_(n=1)^(infty)((-1)^(n-1)x^(2n-1))/((2n-1)!)
= cosx+isinx.

It can also be demonstrated using a complex integral. Let

z = costheta+isintheta
dz = (-sintheta+icostheta)dtheta
= i(costheta+isintheta)dtheta
= izdtheta
int(dz)/z = intidtheta
lnz = itheta,


z = e^(itheta)
= costheta+isintheta.

A mathematical joke asks, How many mathematicians does it take to change a light bulb?" and answers "-e^(ipi)  (which, of course, equals 1.

Also see

0 (zero) It started its life as a space on a counting board. Medieval mathematicians could not decide if it was a number.  It originated in India as an extra numeral.

0.01123595505...  equals 1/89. This is related to the Fibonacci series.  89 is a Fibonacci number and a Fibonacci prime.   The decimal expansion of its reciprocal has the first six Fibonacci numbers: 0,1,1,2,3,5

Think of the Fibonacci series as being a sequence of decimal fractions, arranged so the right most digit of the nth Fibonacci number is in the n+1th decimal place. Then add:
+       .
Proof at

0.1234567891011121314...  The Champernowne constant C10 is a transcendental real constant whose decimal expansion has important properties. It is named after mathematician D. G. Champernowne..

For base 10, the number is defined by concatenating representations of successive integers:

C10 = 0.12345678910111213141516…  Champernowne constants can also be constructed in other bases, similarly, for example:    C2 = 0.11011100101110111… 2       C3 = 0.12101112202122… 3.

The Champernowne constant can be expressed exactly as an infinite series:


and this series generalizes to arbitrary bases b by replacing 10 and 9 with b and b − 1 respectively.

0.20788...  is i to the i-th power.  If you are familiar with complex numbers, the "imaginary" number i has the property that the square of i is -1. It is a rather curious fact that i raised to the i-th power is actually a real number.

From Euler's formula, we know that exp(i*x) = cos(x) + i*sin(x), where "exp(z)" is the exponential function ez. Then

exp(i*Pi/2) = cos(Pi/2) + i*sin(Pi/2) = i.

Raising both sides to i-th power, we see that the right side is the desired quantity ii, while the left side becomes exp(i*i*Pi/2), or exp(-Pi/2), which is approximately .20788.

This is one of many possible values for i to the i, because, for instance, exp(5i*Pi/2) = i. In complex analysis, one learns that exponentiation with respect to i is a multi-valued function.

0.57721566...   The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma (\gamma).

It is defined as the limiting difference between the harmonic series and the natural logarithm:

\gamma = \lim_{n \rightarrow \infty } \left(
\sum_{k=1}^n \frac{1}{k} - \ln(n) \right)=\int_1^\infty\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx.

Here, \lfloor x\rfloor represents the floor function.

The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is


1 (One) The Greeks did not consider 1 to be a number.  It is not defined as a prime number.

The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as  hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.

Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process  indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.

Paul Erdős said, allegedly, about the Collatz conjecture: "Mathematics is not yet ripe for such problems." He also offered $500 for its solution.  Use a calculator at


The one-woman town of Monowi, Nebraska is the only officially incorporated municipality with a population of 1. The sole, 83-year-old resident is the city's mayor, librarian, and bartender. 

1.41421356237309504880168872420969807856967187537694807317667973799... = Square root of 2. 

Binary 1.01101010000010011110…
Decimal 1.4142135623730950488…
Hexadecimal 1.6A09E667F3BCC908B2F…
Continued fraction

The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1It was probably the first number known to be irrational.

The numerical value for the square root of two, truncated to 65 decimal places, is:


    See and

1.5 Is the average of 0 and 1,  See

1.618030339887... This is the Golden Ratio.  Its value is:

 \varphi = \frac{1+\sqrt{5}}{2} = 1.6180339887\ldots.

The Fibonacci numbers are the sequence of numbers {F_n}_(n=1)^infty defined by the linear recurrence equation  F_n=F_(n-1)+F_(n-2) with F_1=F_2=1.  It is conventional to define F_0=0.  The Fibonacci numbers for n=1, 2, ... are 1, 1, 2, 3, 5, 8, 13, 21, ...  

Johannes Kepler discovered that as n increases, the ratio of the successive terms of the Fibonacci sequence {Fn} approaches the golden ratio. .  In 1765, Leonhard Euler published an explicit formula, known today as the Binet formula,

 F_n = {{\varphi^n-(-1/\varphi)^{n}} \over {\sqrt 5}}.

It demonstrates that the Fibonacci numbers grow at an exponential rate equal to the golden ratio φ.

The random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation fn = fn−1 ± fn−2, where the signs + or − are chosen at random with equal probability 1/2, independently for different n. In 1960, Hillel Furstenberg and Harry Kesten showed that for a general class of random matrix products, the norm grows as λn, where n is the number of factors. Their results apply to a broad class of random sequence generating processes that includes the random Fibonacci sequence

In 1999, Divakar Viswanath showed that the growth rate of the random Fibonacci sequence is equal to 1.13198824…, a mathematical constant that was later named Viswanath's constant. As a consequence, the nth root of |fn| converges to a constant value almost surely, or with probability one:

 \sqrt[n]{|f_n|} \to 1.13198824\dots \text{ as } n \to \infty.

2 (Two) raised to the 5th power times 9 raised to the 2nd power = 2592                                               

"A man is a person who will pay two dollars for a one-dollar item he wants. A woman will pay one dollar for a two-dollar item she doesn't want..." -- William Binger

2.0663615...    This is the square root of one half the product of π and e.    Ramanujan showed this equals:

1 + 1/3 + 1/3*5 + 1/3*5*7 + 1/3*5*7*9 + ...

It is not known if this is transcendental.

2.71828....  is approximately equal to e is an important mathematical constant that is the base of the natural logarithm. It is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series:

e =  \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots

In 1737 Euler proved that e is irrational.  In 1873 Hermite proved that e was transcendental.

It is not known if pi+e or pi/e is irrational.

The number e occurs naturally in connection with many problems involving asymptotics. A prominent example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers e and π enter:

n! \sim \sqrt{2\pi n}\, \left(\frac{n}{e}\right)^n.

A particular consequence of this is

e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}.
Euler's formula: e^{ix} = \cos x + i\sin x,\,\!
Euler's formula with x = π is Euler's identity: e^{i\pi} + 1 = 0\,\!
Logarithmic formula: \ln (-1) = i\pi.\,\!
de Moivre's formula.: (\cos x + i\sin x)^n = \left(e^{ix}\right)^n = e^{inx} = \cos (nx) + i \sin (nx),

The letter e was first publicly used to represent the number 2.71828... in a letter from Euler to Goldbach in 1731.

See  for many formulas.


The constant e is base of the natural logarithme is sometimes known as Napier's constant, although its symbol (e) honors Euler.

e is the unique number with the property that the area of the region bounded by the hyperbola y=1/x, the x-axis, and the vertical lines x=1 and x=e is 1. In other words,


With the possible exception of pie is the most important constant in mathematics since it appears in myriad mathematical contexts involving limits and derivatives. The numerical value of e is


e can be defined by the limit


(illustrated above), or by the infinite series


as first published by Newton (1669; reprinted in Whiteside 1968, p. 225).

e is given by the unusual limit


(Brothers and Knox 1998).

Euler (1737; Sandifer 2006) proved that e is irrational by proving that e has an infinite simple continued fraction (e=[2,1,2,1,1,4,1,1,6,...]; Nagell 1951), and Liouville proved in 1844 that e does not satisfy any quadratic equation with integral coefficients (i.e., if it is algebraic, it must be algebraic of degree greater than 2). Hermite subsequently settled the issue, proving e to be transcendental in 1873. However, e is the "least" transcendental possible, with irrationality measure mu(e)=2.

Sondow (2006) proved that e is irrational using a construction for e as the intersection of a nested sequence of closed intervals. This method also provides a measure of irrationality in terms of the Smarandache function (denoted here as S(n) instead of the conventional mu(n) in order to avoid confusion with the irrationality measure) by showing that if p and q are any integers with q>1, then


It is not known if pi+e or pi/e is irrational. It is known that pi+e and pi/e do not satisfy any polynomial equation of degree <=8 with integer coefficients of average size 10^9 (Bailey 1988, Borwein et al. 1989), but it is not known if either of these is transcendental.

It is not known if e is normal to any base (Stoneham 1970).

e has the series representation


as well as

e = [sum_(k=0)^(infty)(1-2k)/((2k)!)]^(-1)
= sum_(k=0)^(infty)(2k+1)/((2k)!)
= 1/2sum_(k=0)^(infty)(k+1)/(k!)
= 2sum_(k=0)^(infty)(k+1)/((2k+1)!)
= sum_(k=0)^(infty)(3-4k^2)/((2k+1)!)
= sum_(k=0)^(infty)((3k)^2+1)/((3k)!)
= [sum_(k=0)^(infty)(4k+3)/(2^(2k+1)(2k+1)!)]^2.

The special case of the Euler formula


with x=pi gives the beautiful identity


an equation connecting the fundamental numbers ipie, 1, and 0 (zero) and involving the fundamental operations of equality (=), addition (+), multiplication (×), and exponentiation.

A nested series for e can be obtained by rewriting the series (2) for e as

e = 1+1+1/(2!)+1/(3!)+1/(4!)+...
= 1+1+1/2(1+1/3+1/(4·3)+...)
= 1+1+1/2(1+1/3(1+1/4(1+1/5(1+...)))),

which gives a pretty nested radical result when x is taken to the power of both sides.

An unexpected Wallis-like formula for e is given by the Pippenger product


(OEIS A084148 and A084149; Pippenger 1980). Another product for e given by


due to Guillera (Sondow 2006). This is analogous to the products




(Guillera and Sondow 2005, Sondow 2006).

Using the recurrence relation


with a_1=a^(-1), compute


2.9951...  The Gelfond–Schneider constant or Hilbert number is


which was proved to be a transcendental number by Rodion Kuzmin in 1930. In 1934, Aleksandr Gelfond proved the more general Gelfond–Schneider theorem, which solved the part of Hilbert's seventh problem described below.

Part of the seventh of Hilbert's twenty three problems posed in 1900 was to prove (or find a counterexample to the claim) that ab is always transcendental for algebraic a ≠ 0, 1 and irrational algebraic b. In the address he gave two explicit examples, one of them being the Gelfond–Schneider constant 2√2.

In 1919, he gave a lecture on number theory and spoke of three conjectures: the Riemann hypothesis, Fermat's Last Theorem, and the transcendence of 2√2. He mentioned to the audience that he didn't expect anyone in the hall to live long enough to see a proof of this final result. But the proof of this number's transcendence was published by Kuzmin in 1930, well within Hilbert's own lifetime. Namely, Kuzmin proved the case where the exponent b is a real quadratic irrational, which was later extended to an arbitrary algebraic irrational b by Gelfond.

3 (Three) The only three consecutive integers whose cubes sum to a cube are given by the Diophantine equation:

3 cubed + 4 cubed + 5 cubed = 6 cubed.

"Cubes":  153, 370, 371 and 407 are all the "sum of the cubes of their digits". In other words 153=13+53+33

A circle can be drawn through three points that are not colinear.

3.14159...   called π,  is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. It is an irrational number. Like all irrational numbers, π cannot be represented as a common fraction.

Every irrational number, including π, can be represented by an infinite series of nested fractions, called a continued fraction:

\pi=3+\textstyle \frac{1}{7+\textstyle \frac{1}{15+\textstyle \frac{1}{1+\textstyle \frac{1}{292+\textstyle \frac{1}{1+\textstyle \frac{1}{1+\textstyle \frac{1}{1+\ddots}}}}}}}

When Euler solved the Basel problem (See in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between π and the prime numbers that later contributed to the development and study of the Riemann zeta function:

 \frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots

Swiss scientist Johann Heinrich Lambert in 1761 proved that π is irrational, meaning it is not equal to the quotient of any two whole number Lambert's proof exploited a continued-fraction representation of the tangent function. French mathematician Adrien-Marie Legendre proved in 1794 that π2 is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π is transcendental, confirming a conjecture made by both Legendre and Euler.

Other series involving π are found at:  Here a very few:

pi/4 = sum_(k=1)^(infty)((-1)^(k+1))/(2k-1) = 1-1/3+1/5-...   discovered by Leibnitz in 1673.



1/4pisqrt(2) = sum_(k=1)^(infty)[((-1)^(k+1))/(4k-1)+((-1)^(k+1))/(4k-3)] = 1+1/3-1/5-1/7+1/9+1/(11)-...

1/4(pi-3) = sum_(k=1)^(infty)((-1)^(k+1))/(2k(2k+1)(2k+2)) = 1/(2·3·4)-1/(4·5·6)+1/(6·7·8)-...

1/6pi^2 = sum_(k=1)^(infty)1/(k^2)= 1+1/4+1/9+1/(16)+1/(25)+...

1/8pi^2 = sum_(k=1)^(infty)1/((2k-1)^2)= 1+1/(3^2)+1/(5^2)+1/(7^2)+...

pi = (3sqrt(3))/4+24(1/(12)-1/(5·2^5)-1/(28·2^7)-1/(72·2^9)-...), was determined by Isaac Newton in 1666.

pi/2 =1/2sum_(n=0)^(infty)((n!)^22^(n+1))/((2n+1)!)=sum_(n=0)^(infty)(n!)/((2n+1)!!) = 1+1/3+(1·2)/(3·5)+(1·2·3)/(3·5·7)+... = 1+1/3(1+2/5(1+3/7(1+4/9(1+...))))


 pi/(5sqrt(phi+2))=1/2sum_(i=0)^infty((i!)^2)/(phi^(2i+1)(2i+1)!), where phi is the golden ratio.



The first 16 million digits of pi have passed all the tests of being random.

Also look at and

Bellard's formula, as used by PiHex, the now-completed distributed computing project, is used to calculate the nth digit of π in base 2. It is a faster version (about 43% faster[1]) of the Bailey–Borwein–Plouffe formula.  It was discovered by Fabrice Bellard in 1997.


Pi is also a transcendental number, which means it cannot be the root of an algebraic equation.  

The probability that an integer does not contain any squared factors, and the probability that two numbers chosen at random are relatively prime is  π2/6 is about 61%.

If ζ refers to the Riemann zeta function, the identity relating the product over primes to ζ(2) is an example of an Euler product, and the evaluation of ζ(2) as π2/6 is the Basel problem, solved by Leonhard Euler in 1735.

More generally, the probability of k randomly chosen integers being coprime is 1/ζ(k).

Relatively Prime

DOWNLOAD Mathematica Notebook EXPLORE THIS TOPIC IN the MathWorld Classroom Contribute to this entry Relatively prime plots

Two integers are relatively prime if they share no common positive factors (divisors) except 1. Using the notation (m,n) to denote the greatest common divisor, two integers m and n are relatively prime if (m,n)=1. Relatively prime integers are sometimes also called strangers or coprime and are denoted m_|_n. The plot above plots m and n along the two axes and colors a square black if (m,n)=1 and white otherwise (left figure) and simply colored according to (m,n) (right figure).

Two numbers can be tested to see if they are relatively prime in the Wolfram Language using CoprimeQ[m, n].

Two distinct primes p and q are always relatively prime, (p,q)=1, as are any positive integer powers of distinct primes p and q, (p^m,q^n)=1.

Relative primality is not transitive. For example, (2,3)=1 and (3,4)=1, but (2,4)=2.

The probability that two integers m and n picked at random are relatively prime is


(OEIS A059956; Cesàro and Sylvester 1883; Lehmer 1900; Sylvester 1909; Nymann 1972; Wells 1986, p. 28; Borwein and Bailey 2003, p. 139; Havil 2003, pp. 40 and 65; Moree 2005), where zeta(z) is the Riemann zeta function. This result is related to the fact that the greatest common divisor of m and n, (m,n)=k, can be interpreted as the number of lattice points in the plane which lie on the straight line connecting the vectors (0,0) and (m,n) (excluding (m,n) itself). In fact, 6/pi^2 is the fractional number of lattice points visible from the origin (Castellanos 1988, pp. 155-156).

Given three integers (k,m,n) chosen at random, the probability that no common factor will divide them all is


(OEIS A088453; Wells 1986, p. 29), where zeta(3) is Apéry's constant (Wells 1986, p. 29). In general, the probability that n random numbers lack a pth power common divisor is [zeta(np)]^(-1) (Cohen 1959, Salamin 1972, Nymann 1975, Schoenfeld 1976, Porubský 1981, Chidambaraswamy and Sitaramachandra Rao 1987, Hafner et al. 1993).

Interestingly, the probability that two Gaussian integers a and b are relatively prime is


(OEIS A088454), where K is Catalan's constant (Pegg; Collins and Johnson 1989; Finch 2003, p. 601).

Similarly, the probability that two random Eisenstein integers are relatively prime is


(OEIS A088467), where


(Finch 2003, p. 601), which can be written analytically as

H = 1/9[psi_1(1/3)-psi_1(2/3)]
= 0.78130...

 where psi_1(z) is the trigamma function

Amazingly, the probabilities for random pairs of integers and Gaussian integers being relatively prime are the same as the asymptotic densities of squarefree integers of these types.

See and

The Feynman point is a sequence of six 9s that begins at the 762nd decimal place of the decimal representation of π. It is named after physicist Richard Feynman, who once stated during a lecture he would like to memorize the digits of π until that point, so he could recite them and quip "nine nine nine nine nine nine and so on", suggesting, in a tongue-in-cheek manner, that π is rational.

Which is a better fit: a round peg in a square hole or a square peg in a round hole?  The ratios are:  π/2  and 2/π .As π squared is greater than 8, the round peg is a better fit and fills more of the square hole.

"Pi Day" is celebrated on March 14.  "Feigenbaum Constant Day" is celebrated on April 6 (or April 7 if you round up)

Buffon's Needle Problem involves pi. . See and additional formulas by Euler  and  note the famous formula for     pi /4 equals

(3x5x7x11x13x17x19x ...)/(4x4x8x12x12x16 ,,,)  where the numerator consists of the odd prime numbers and the numbers in the  denominator are multiples of 4 that are closest to the corresponding number in the numerator.

It can shown that Pi is irrational. Also see  However, John Arioni's simple proof of the infinitude of primes uses the fact that Pi/4 can be expressed as a product of two series each involving primes. John observed that the identity implies infinitude of primes, because had the number of primes been finite, the right-hand side would have been rational, while the left side Pi/4 is known to be irrational. 

  • John Wallis (1655) took what can now be expressed as


    INTEGRAL{0...1}Squareroot[1-x^2]dx = pi/4

    and without using the binomial theorem or integration (not invented yet) painstakingly came up with a formula for pi to be


    pi = 2*(2*2*4*4*6*6*...)/(1*3*3*5*5*7*...) .


  • William Brouncker (ca. 1660's) rewrote Wallis' formula as a continued fraction, which Wallis and later Euler (1775) proved to be equivalent. It is unknown how Brouncker himself came up with the continued fraction,


    4/pi = 1 + (1^2)/(2 + (3^2)/(2 + (5^2)/(2 + (7^2)/(2 + (9^2)/(2 + ... ))))) .


  • James Gregory (1671) & Gottfried Leibniz (1674) used the series expansion of the arctangent function,


    arctan(x) = x - (x^3)/3 + (x^5)/5 - (x^7)/7 + (x^9)/9 - (x^11)/11 + ... ,

    and the fact that arctan(1) = pi/4 to obtain the series


    pi = 4*(1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ... .

    Unfortunately, this series converges to slowly to be useful, as it takes over 300 terms to obtain a 2 decimal place precision. To obtain 100 decimal places of pi, one would need to use at least 10^50 terms of this expansion!


  • History books credit Sir Isaac Newton (ca. 1730's) with using the series expansion of the arcsine function,


    arcsin(x) = x + (1*x^3)/(2*3) + (1*3*x^5)/(2*4*5) + (1*3*5*x^7)/(2*4*6*7) + ... ,

    and the fact that arctan(1/2) = pi/6 to obtain the series


    pi = 6*(1/2 + 1/(2*3*2^3) + 1/5 - 1/7 + 1/9 - 1/11 + ...) .

    This arcsine series converges much faster than using the arctangent. (Actually, Newton used a slightly different expansion in his original text.) This expansion only needed 22 terms to obtain 16 decimal places for pi.


  • Leonard Euler (1748) proved the following equivalent relations for the square of pi,




  • Ko Hayashi (1989) found another infinite expression for pi in terms of the Fibonacci numbers,




    4   One half of five is four, as one half of five is iv.  Write any number from 1 to 100 using four 4's:    (See A hyp  erbola can be drawn though four points, three of which are not colinear. In 1878 Francis Guthrie said a map could be colored with at most  four colors so no abutting country had the same color. This 4 color conjecture has been "proven" correct nontraditionally using a computer. Every integer can be represented as the sum of at most four squares.

    4.11325037878... This is e raised to the square root of 2.

    4.6692...  Some mathematical mappings involving a single linear parameter exhibit the apparently random behavior known as chaos when the parameter lies within certain ranges. As the parameter is increased towards this region, the mapping undergoes bifurcations at precise values of the parameter. At first there is one stable point, then bifurcating to an oscillation between two values, then bifurcating again to oscillate between four values and so on. In 1975, Dr. Feigenbaum, using the small calculator he had been issued, discovered that the ratio of the difference between the values at which such successive period-doubling bifurcations occur tends to a constant of around 4.6692... He was able to provide a mathematical proof of that fact, and he then showed that the same behavior, with the same mathematical constant, would occur within a wide class of mathematical functions, prior to the onset of chaos. For the first time, this universal result enabled mathematicians to take their first steps to unraveling the apparently intractable "random" behavior of chaotic systems. This "ratio of convergence" is now known as the first Feigenbaum constant which equals 4.6692... 

    And the ratio between the values where the period doubles ends up approaching the Feigenbaum constant, approximately 4.669.

    So, should we celebrate Feigenbaum Constant Day on April 6th or 7th? Due to rounding, it's probably the 7th, but feel free to celebrate both.

    Bifurcation diagram of the logistic map below. Feigenbaum noticed in 1975 that the quotient of successive distances between bifurcation events tends to 4.6692...  This is the first Feigenbaum constant.  The second is 2.5029.... described later.  See,, and especially

    The first Feigenbaum constant is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map

    x_{i+1} = f(x_i)

    where f(x) is a function parameterized by the bifurcation parameter a.

    It is given by the limit:

    \delta = \lim_{n\rightarrow \infty} \dfrac{a_{n-1}-a_{n-2}}{a_n-a_{n-1}} = 4.669\,201\,609\,\cdots

    where an are discrete values of a at the nth period doubling.


    We can make some extraordinary observations from the bifurcation diagram. The ratio between each bifurcation converges to the first Feigenbaum’s constant, a universal constant for functions that have a periodic-doubling route to chaos, a fact discovered by Mitchell Feigenbaum in 1975. This constant is approximately 4.669...  .. We can interpret from this constant that as we approach chaos each periodic region is smaller than the previous region by a factor of 4.669 (the first Feigenbaum constant).

    The extraordinary thing about this constant is that its the same for all uni-modal functions (functions having a quadratic maximum and approach chaos via period doubling). If we call r_n the value of r for which the 2^n period becomes unstable then \delta = \lim_{n\to\infty}\frac{r_{n+1}-r_n}{r_{n+2}-r_{n+1}}=4.669....

    2.5029...   is the second Feigenbaum constant.  The following functional equation also arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade. The functional equation is the mathematical expression of the universality of period doubling. The equation is used to specify a function g and a parameter λ by the relation

     g(x) = \frac{1}{-\lambda} g( g(\lambda x ) )

    with the boundary conditions

    bulletg(0) = 1,
    bulletg′(0) = 0, and
    bulletg′′(0) < 0

    For a particular form of solution with a quadratic dependence of the solution near x=0, the inverse 1/λ=2.5029... is the second Feigenbaum constant. In bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the mathematician Mitchell Feigenbaum.

    The second Feigenbaum constant     \alpha = 2.502907875095892822283902873218...,  is the ratio between the width of a tine and the width of one of its two subtines (except the tine closest to the fold). A negative sign applied to \alpha when the ratio between the lower subtine and the width of the tine is measured.  (In chaos theory ( non-linear dynamics), the branches of a bifurcation diagram are called tines and subtines.)

    The Feigenbaum constants apply to a large class of dynamical systems.  Both constants are believed to be transcendental

    5  Five is the first prime of the form 6n - 1.  In fact all primes, except 2 and 3, are one more or less than a multipe of 6.  Pappus showed how to construct a conic through any five points in the plane, no three of which are colinear.

    Is also a pentagonal number. A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside.

    pn is given by the formula:  (n)(3n - 1)/2

    for n ≥ 1. The first few pentagonal numbers are:

    15122235517092117145176210247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187..

      The problem of finding two rational numbers whose cubes sum to six was "proved" impossible by Legendre. However, Dudeney found the simple solutions 17/21 and 37/21. Did you know that snowflakes have six sides.  Six equals the square root of the sum of the cubes of 1 and 2 and 3.  There are 6 regular polytopes.  They are analogous in four dimensions of the regular polyhedra in three dimensions and the regular polygons in two dimensions.

    7  Did you know the 10 millionth digit of π is a 7?  So what!

    Seven is a prime.  See the following table by Martin Gardner, which contains all primes.  Start with the top one and peel off right hand digits.

    The "Flower of Life"is a name for a geometrical figure composed of multiple evenly-spaced, overlapping circles. This figure, used as a decorative motif since ancient times, forms a flower-like pattern with the symmetrical structure of a hexagon.  A "Flower of Life" figure consists of seven overlapping circles, in which the center of each circle is on the circumference of up to six surrounding circles of the same diameter. However, the surrounding circles need not be clearly or completely drawn; in fact, some ancient symbols that are claimed as examples of the Flower of Life contain only a single circle or hexagon. 

    The "Seed of Life" is formed from seven circles being placed with sixfold symmetry, forming a pattern of circles and lenses, which act as a basic component of the Flower of Life's design.

    8.53973422267... This is π times e.   Most sums, products, powers of the number π and the number e are not known to be rational, algebraic irrational or transcendental ( π + e, π − e, πe, π/e, ππ, ee, πe, π2, eπ2 ).  However, π + eπ, πeπ and eπ√n ,for any positive integer n, have been proven to be transcendental.

    9   Dudeney found two rational numbers other than 1 and 2 whose cubes sum to nine:

     [415280564497 / 348671682660]  and [676702467503 / 348671682660]

    Nine is the maximum number of cubes that are needed to sum to any positive integer.  Nineteen is the maximum number of fourth powers needed to sum to any positive integer.

    A number is divisible by 9 if, and only if, if it divides the sum of the number's digits.

    9.8960440189358 ...  This is π squared.  In 1794, Legendre proved this was irrational.

    10   The base of the decimal system.   10! =6! x 7!  These are the only consecutive integers, 6 and 7, that solve the equation N! = A! x (A+1)!

    11  British mathematician J J Sylvester said: "Mathematics is the music of reason."  In 1884 at age 70, he proved that the highest number that cannot be created from using two numbers x and y equals xy - x - 7.  In Rugby, where drop goals are scored as 3 and converted tries are scored as 7, there cannot ever be a score of 11.            That is 3x7 -  3 - 7 = 11. 

     11 is a palindrome, a number that can be read forward and backward.  But so are some of their squares.           

       Dattaraya Ramchandra Kaprekar (1905–1986) was an Indian recreational mathematician who described several classes of natural numbers including the Kaprekar, Harshad and Self numbers and discovered the Kaprekar constant, named after him. Despite having no formal postgraduate training and working as a schoolteacher, he published extensively and became well known in recreational mathematics circles. .International fame arrived when Martin Gardner wrote about Kaprekar in his March 1975 column of Mathematical Games for Scientific American. Today his name is well-known and many other mathematicians have pursued the study of the properties he discovered. 

    Kaprekar also studied the Demlo numbers, named after a train station 30 miles from Bombay on the then G. I. P. Railway where he had the idea of studying them. These are the numbers 1, 121, 12321, …, which are the squares of the repunits 1, 11, 111,

    11 is the second repunit, a number whose digits are all 1s. This works for n =2 and for some higher n's:

    n 11 to the nth 111 to the nth 1111 to the nth 11111 to the nth 111111 to the nth
    1 11 111 1111 11111 111111
    2 121 12321 1234321 123454321 12345654321
    3 1331 1367631      
    4 14641        

    12  The base of the duodecimal system.  The Romans used only duodecimal fractions.  The dodecahedron has 12 faces.

    13  The next number in the sequence 345 is 1. The next number in the sequence 543 is 1.  The next number in the sequence 222 is 7.  The next number in the sequence 123 is 7.  The next number in the sequence 333 is 4. These are all bridge hand distributions.

    "Triskaidekaphobia" is the fear of Friday the 13th.  Gauss developed a calendar formula to determine the day of the week upon which any day of the week falls.  In a 400 year cycle, there are 12 months that have a 13th, or 4800 days. The average you might expect on any day of the week is 4800/7 = 685.7.  But Friday is the most frequent at 688, followed by Wed and Sun at 687, Mon and Tue at 685,  and Thur and Sat at 684.

    13 is supposedly unlucky: 13 present at Last Supper, 13 at a banquet in Bahalla when Balder (don of Odin) was slain. Friday is not a good day: Eve tempted Adam to eat the forbidden fruit, the Great Flood occurred, the start of linguistic confusion at the Tower of Babel, the destruction of Solomon's Temple, the death of Jesus Christ, British sailors fear of sailing on a Friday.

    14.134725 ... The first zero in the Reiman zeta function is at 1/2 + i(14.134725 ...  )

    17  A 17 sided regular polygon is a Heptadecagon.In 1796, when Gauss was only 19 he announced he could construct this with straight edge and compass.  He went further and indicated which n-gons could be constructed using Euclidean tools and which could not. 17 is equal to the um of the digits of its cube, 4913.  It is the only prime number to have this property.  Also, the formula, n squared + n + 17, generates prime numbers for n = 0 to 15.

    19  Every single positive integer can be written as the sum of at most 19 powers.

    22, 23, and 24 are the only positive integers (other than 1) for which n! has precisely n digits.

    22.4591577183 ... This is pi raised to the e power, πe.  It is not known if this is irrational or transcendental.

    23  Its  digits are consecutive prime numbers.  It is the smallest odd prime that is not a twin prime.  23 is the smallest prime for which the sum of the squares of its digits is also a prime.  There were 23 problems on David Hilbert's famous list of unsolved mathematical problems,  Beckham chose the number 23 on his shirt to play for Real Madrid. Michael Jordan wore the number 23.  We have 23 pairs of chromosomes.   Caesar was stabbed 23 times. 23 mph is the maximum speed of an American crow. 

    23 equals 3x5 + 3 +5

    23 factorial contains 23 digits, the only prime number with this property.

     23.1406926328... is e raised to the power pi.  In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is eπ, that is, e to the power of π. Like both e and π, this constant is a transcendental number. This was first established by Gelfond and may now be considered as an application of the Gelfond–Schneider theorem, noting the fact that

     e^\pi = (e^{i\pi})^{-i} = (-1)^{-i},

    where i is the imaginary unit. Since −i is algebraic but not rational, eπ is transcendental. The constant was mentioned in Hilbert's seventh problem.  A related constant is 2^\sqrt{2}, known as the Gelfond–Schneider constant. The related value π + eπ is also irrational.

    24 is a factor of the product of 4 consecutive integers.  The product of n consecutive integers is divisible by n factorial !

    25  is a Friedman Number: A  positive integer which can be written in some non-trivial way using its own digits, together with the symbols + - x / ^ ( ) and concatenation. See:       Here, 25 = 52  

    31 is a Mersenne  prime number. That is, it is a prime number that can be written in the form M(n) = 2ⁿ − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. The first four Mersenne primes are 3, 7, 31, and 127

    31 is the number of regions a circle can be divided into by connecting 6 points on the edge of a circle.  See and 

    The surprising result is not 32 as the pattern of the number of regions seems to be following a power of 2.

    Points 1 2 3 4 5 7 8
    Regions 1 2 4 8 16 31 57 99
    If you have a circle with n points on the circle.

    You can connect lines connecting these points, where the number of lines = L(n)= (n)(n-1)/2

    These number of intersections of these lines  = I(n) = (n)(n-1)(n-2)(n-3)/24

    The number of regions formed by these lines and their intersections = R(n) = 1 + L(n) + I(n)

    R(n) = 1 + (n)(n-1)/2  + (n)(n-1)(n-2)(n-3)/24

    31 is the number of regular polygons with an odd number of sides that are known to be constructible with compass and straightedge.

    37   (666)/(6 + 6 + 6) = 111/3

    Forty is the only number whose letters are in alphabetical order.

    45  Lake Chaubunagungamaug, also known as Webster Lake, is a lake in the town of Webster, Massachusetts, United States. It is located near the Connecticut border and has a surface area of 1442 acres. Since 1921, the lake has also been known by a much longer name having 45 letters comprising fourteen syllables: Lake Char­gogg­a­gogg­man­chaugg­a­gogg­chau­bun­a­gung­a­maugg.

    53  If there are 53 people in a room, the probability is approximately 1/53 none of them will share a birthday.  This is .019...  ,  about 2%.

    55 is the largest two digit number used in the NBA.  This makes it easy for referees to communicate a player's number using hand signals using the fingers on each hand. Each digit is 0 thru 5.

    41  Euler discovered the formula n squared + n + 41 generates prime numbers for n equal to 0 to 39.

    60  The base of the sexagesimal system used by the Sumerians as early as 3500 BC.  The Babylonians used this system for mathematical and astronomimal work. The division of the circle into 360 degrees and the degrees into 60 parts originated with the Babylonians.  We still do this with time, measured in 60 minutes and 60 seconds.

    61  The smallest solution of the Pellian equation x squared minus 61 times y squared = 1, is x = 1,766,319,049 and y = 226,153,980.  Wow!

    71  71 squared = 7 factorial + 1 factorial.  That is, the sum of the factorials of its digits.

    88  The number of keys on a piano, 52 white keys and 36 black keys. . There are 7 white keys and 5 black keys to an octave.    88  The number that is called "two fat ladies" in Bingo. 88  The number of feet  per second, when driving 60 miles per hour.

    93  You can chop a big lump of cheese into a maximum of 93 bits with 8 straight cuts.

    100  The square of 10, the base of the decimal system, but also the square of the base in any other base.

    113  The smallest three digit prime such that all other arrangements of its digits are also prime (131 and 311). Other three digit numbers are 199 (919 and 991) and 337 (373 and 733).  Two digit primes are 12, 13, 17, 37 and 79.  The next two primes with this property are 1,111,111,111,111, 111,111 and a similar prime consisting of 23 instead of 19 1s.

    128  A cord of wood is 4 feet by 4 feet by 8 feet or 128 cubic feet. A tennis tournament with 2n  players will have n rounds. A Grand slam Singles Tournament has 128 entrants and 7 rounds. 128 is the largest number that cannot be expressed as the sum of three distinct squares. Like 25, 128 is another Friedman number: 128 = 28-1

    129  It can be expressed as the sum of three distinct squares in two different ways.  129 = 100 + 25 +4.  Also 129 = 64 + 49 + 16.

    144 It is the largest Fibonacci square.

    153  It is the sum of the first 17 integers.  .

    It is also the sum of the first five positive factorials:1!+2!+3!+4!+5!

    It is also the smallest three-digit number which can be expressed as the sum of cubes of its digits.  153 = 1^3 + 5^3 + 3^3 All such numbers below 10 to the tenth power are: 0, 1, 153, 370, 371, 407, 165033, 221859, 336700, 336701, 340067, 341067, 407000, 407001, 444664, 487215, 982827, 983221, 166500333, 296584415, 333667000, 333667001, 334000667, 710656413, 828538472.

    216  It is the smallest cube that is the sum of three cubes:  216 = 3^3 + 4^3 + 5^3 = 6^3

    365   It is the  smallest number which has more than one expression as a sum of consecutive squares.

    365 = 13^2 + 14^2         365 = 10^2 + 11^2 + 12^2
    538  is the number of electors in the U S Electoral College.  It is also the name of Nate Silver's website, purchased by ESPN in July 2013, that focuses on opinion poll analysis, politics, economics, and sports blogging.  The website's logo is a fox, in reference to a phrase attributed to Archilochus: "The fox knows many things, but the hedgehog knows one big thing."   
    Scholars have differed about the correct interpretation of these dark words, which may mean no more than that the fox, for all his cunning, is defeated by the hedgehog's one defense. But, taken figuratively, the words can be made to yield a sense in which they mark one of the deepest differences which divide writers and thinkers, and, it may be, human beings in general. For there exists a great chasm between those, on one side, who relate everything to a single central vision which has significance-and, on the other side, those who pursue many ends, often unrelated and even contradictory, connected, if at all, only in some de facto way, for some psychological or physiological cause, related by no moral or aesthetic principle.
    In The Hedgehog and the Fox, an essay by philosopher Isaiah Berlin, Berlin expands upon an idea to divide writers and thinkers into two categories: hedgehogs, who view the world through the lens of a single defining idea (examples given include Plato, Lucretius, Dante Alighieri, Blaise Pascal, Georg Wilhelm Friedrich Hegel, Fyodor Dostoyevsky, Friedrich Nietzsche, Henrik Ibsen, Marcel Proust and Fernand Braudel), and foxes, who draw on a wide variety of experiences and for whom the world cannot be boiled down to a single idea (examples given include Herodotus, Aristotle, Desiderius Erasmus, William Shakespeare, Michel de Montaigne, Molière, Johann Wolfgang Goethe, Aleksandr Pushkin, Honoré de Balzac, James Joyce and Philip Warren Anderson).
    563  In number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if

    (n-1)!\ \equiv\ -1 \pmod n.

    Occasionally  it is divisible by p squared.  In 1953, using a computer, Goldberg found that the only value for of  n = p were 5,  13, and 563.

    640 are the number of acres in one square mile.

    666  is the occult "number of the beast," also called the "sign of the devil", associated in the Bible with the Antichrist. It has figured in many numerological studies. It is mentioned in Revelation 13:18: "Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is 666."

    The beast number has several interesting properties which numerologists may find particularly interesting (Keith 1982-83). In particular, the beast number is equal to the sum of the squares of the first 7 primes


    Emanouilidis (1998) also gives additional more obscure connections between 666 and the numbers on a roulette wheel.

    The number 666 is a sum and difference of the first three 6th powers,


    666  is the sum of numbers on a roulette wheel, which 38 pockets labeled 0, 00, and 1-36.

    666 is the sum of the first 144 decimal digits in pi.

    714  The record  number of home runs hit by Babe Ruth, and surpassed by Hank Aaron's 756 and Barry Bonds' 762.  The product of 714 and 715 equals 2x3x5x7x11x13x17.

    720  720 = 10x9x8 = 6x5x4x3x2.

    1000  The smallest number spelled out that has an "a" in it.

    1001 450 factorial has 1001 digits and is appropriately known as the Arabian Nights Factorial.

    In The Thousand and One Arabian  Nights, a king  found out that his first wife was unfaithful to him.  He then married a new virgin each day who would be beheaded the following morning.  Scheherazade, the vizier's daughter, volunteered to be his next wife.  She had perused the books, annals and legends of preceding Kings, and the stories, examples and instances of bygone men and things; indeed it was said that she had collected a thousand books of histories relating to antique races and departed rulers. She had perused the works of the poets and knew them by heart; she had studied philosophy and the sciences, arts and accomplishments; and she was pleasant and polite, wise and witty, well read and well bred.  Once in the king's chambers, Scheherazade began a telling a story. The king lay awake and listened with awe as Scheherazade told her first story. The night passed by, and Scheherazade stopped in the middle of the story. The king asked her to finish, but Scheherazade said there was not time, as dawn was breaking. So, the king spared her life for one day to finish the story the next night. So the next night, Scheherazade finished the story and then began a second, even more exciting tale which she again stopped halfway through at dawn. So the king again spared her life for one day to finish the second story.  And so the King kept Scheherazade alive day by day, as he eagerly anticipated the finishing of last night's story. At the end of 1,001 nights, and 1,000 stories, Scheherazade told the king that she had no more tales to tell him. During these 1,001 nights, the king had fallen in love with Scheherazade, and so he spared her life, and made her his queen.

    1233  Bhavaraju Sarveswara Rao (1915 – 2010) was an Indian economist and a social scientist, who found that    1233 = 12² + 33² .  Also 8833 88² + 33² .  These have four digits.

    The number with six digits is: 990100 =  990² + 100² .

    Numbers with ten digits are:  1765038125 = 17650² + 38125² ,   2584043776 =  25840² + 43776² ,   7416043776 =  74160² + 43776² ,  8235038125 =  82350² + 38125²

    1477  is a factorial prime: a prime number that is one less or one more than a factorial.. The first few factorial primes are:  2 (0! + 1 or 1! + 1), 3 (2! + 1), 5 (3! − 1), 7 (3! + 1), 23 (4! − 1), 719 (6! − 1), 5039 (7! − 1), 39916801 (11! + 1), 479001599 (12! − 1), 87178291199 (14! − 1),

    n! − 1 is prime for:  n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040, 147855, ...

    n! + 1 is prime for:  n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, ...

    No other factorial primes are known as of September 2013.

    1729  is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a visit to the hospital to see the Indian mathematician Srinivasa Ramanujan. In Hardy's words:  I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."  The two different ways are these:   1729 = 13 + 123 = 93 + 103

    1729 is a Harshad number (or Niven number): an integer that is divisible by the sum of its digits. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The term “Niven number” arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977.

    3320  The Gregorian calendar has 365 days, except those years divisible by 400, which have 366 days.  After 3320 days, this will require a downward adjustment of 1.

    4830  is the number of square yards in an acre.

    8902 are the number of ways to play the first four moves in chess.

    26,861  There are as many primes of the form 4n + 1 as of the form 4n + 3 below 26861.  26861 is a prime of the form 4n + 1.  Littlewood proved the lead keeps changing an infinite number of times.

    43,470 is the number of square feet in an acre.

    150, 209 factorial plus 1.  This form of a factorial plus or minus one, that equals a prime number,  is shown below.

    rank prime digits when
    1 150209! + 1 712355 Oct 2011
    2 147855! - 1 700177 Sep 2013
    3 110059! + 1 507082 Jun 2011
    4 103040! - 1 471794 Dec 2010
    5 94550! - 1 429390 Oct 2010
    6 34790! - 1 142891 May 2002
    7 26951! + 1 107707 May 2002
    8 21480! - 1 83727 Sep 2001
    9 6917! - 1 23560 Oct 1998
    10 6380! + 1 21507 Oct 1998
    11 3610! - 1 11277 Oct 1993
    12 3507! - 1 10912 Oct 1992
    13 1963! - 1 5614 Oct 1992
    14 1477! + 1 4042 Dec 1984
    15 974! - 1 2490 Oct 1992
    16 872! + 1 2188 Dec 1983
    17 546! - 1 1260 Oct 1992
    18 469! - 1 1051 Dec 1981

    23,456,789 is a prime with consecutive digits. All that are known are: 23, 67, 89, 4567, 78901, 23456789, 45678901, 9012345678901, 789012345678901.

    The symmetrical decomposition of its digits yields three primes: 23, 4567, and 89.

    73,939,133  is the largest prime, which when successive digits are removed from the right the results are all prime.  Another example is 33,333,331.

    73939133 7393913            739391     73939       7393          739               73                     7

    272,400, 600 are the number of terms it takes for the partial sum of the slowly diverging Harmonic Series to reach 20.

    635,318, 657   See and the Indian mathematician Srinivasa Ramanujan

    1,000,000,000 One billion.  A billion is 109  (2 + 1)x3 = 9 zeros

    1,234,567,891 is one of three primes whose  digits are in ascending order.  The other two are 12,345,678,901,234,567,891 and 1,234,567,891,234,567,891,234,567,891.

    1,480,028,171 is the central prime in a magic square.  Harry Nelson won $100 from Martin Gardner for discovering a 3x3 magic square of primes the other 8 primes are more or less than it by 12, 18, 30, and 42.  Harry Lewis Nelson (born January 8, 1932) is an American mathematician and computer programmer. He was a member of the team that won the World Computer Chess Championship in 1983 and 1986.  He was a co-discoverer of the 27th Mersenne prime in 1979 (at the time, the largest known prime number). He also served as editor of the Journal of Recreational Mathematics for five years.

    1,979,339,339 and its tail cuttings are all primes:










    158,753,389,900:1  The odds against a person being dealt a single suit in bridge.  158 billion to 1. A billion is 109  (2 + 1)x3 = 9 zeros

    608,981,813,029 is the number where primes of the form 3n + 1 become the majority. Prime numbers of the form 3n + 1 are more numerous than those of the form 3n + 2 at 608981813029, even though the "race" initially favors 3n + 2.  The lead then keeps changing hands, but for an infinite number of primes, this 3n + 1 form has the majority. This is over 608 billion.  A billion is 109  (2 + 1)x3 = 9 zeros

    635,013,559,600 are the number of possible bridge hands a player can receive. This is over 635 billion.  A billion is 109  (2 + 1)x3 = 9 zeros

    13! x 39!

    18,446,744,073,709,551,615 This  number 264 - 1 exceeds 18 quintillion.  A quintillion is 1018  (5 +1)x3 = 18 zeros  It appears at least twice in mathematical folklore.

    Chessboard There are different stories about the invention of chess. One of them includes the geometric progression problem. Its earliest written record is contained in the Shahnameh, an epic poem written by the Persian poet Ferdowsi between c. 977 and 1010 CE.  When the creator of the game of chess, an ancient Indian Brahmin mathematician named Sessa or Sissa, showed his invention to the ruler of the country, the ruler was so pleased that he gave the inventor the right to name his prize for the invention. The man, who was very clever, asked the king that for the first square of the chess board, he would receive one grain of wheat, two for the second one, four on the third one, and so forth, doubling the amount each time. The ruler quickly accepted the inventor's offer, even offended by his notion that the inventor was asking for such a low price, and ordered the treasurer to count and hand over the wheat to the inventor. However, when the treasurer took more than a week to calculate the amount of wheat, the ruler asked him for a reason for his tardiness. The treasurer then gave him the result of the calculation, and explained that it would take more than all the assets of the kingdom to give the inventor the reward. The story ends with the inventor becoming the new king.

    Tower of Hanoi In the great temple at Benares beneath the dome that marks the centre of the world, rests a brass plate in which are fixed three diamond needles, each a cubit high and as thick as the body of a bee. On one of these needles, at the creation, God placed sixty-four discs of pure gold, the largest disk resting on the brass plate, and the others getting smaller and smaller up to the top one. This is the tower of Bramah. Day and night unceasingly the priest transfer the discs from one diamond needle to another according to the fixed and immutable laws of Bramah, which require that the priest on duty must not move more than one disc at a time and that he must place this disc on a needle so that there is no smaller disc below it. When the sixty-four discs shall have been thus transferred from the needle which at creation God placed them, to one of the other needles, tower, temple, and Brahmins alike will crumble into dust and with a thunderclap the world will vanish.  The number of separate transfers of single discs which the Brahmins must make to effect the transfer of the tower is two raised to the sixty-fourth power minus 1 or 18,446,744,073,709,551,615 moves. Even if the priests move one disk every second, it would take more than 500 billion years to relocate the initial tower of 64 disks.

    42,252,003,274,489,856,000  the total possible positions that can be reached on the original 3x3x3 Rubic's Cube.  It can be solved in just 20 moves.  There are 6 colors on a Rubik's Cube:  yellow, blue, green, red, white, and orange.  This number exceeds 42 quintillion.  A quintillion is 1018  (5 +1)x3 = 18 zeros

    number of permutations Mathematics of the Rubiks Cube

    147,573,952,589,676,412,927  was proven to not be a Mersenne prime. This prime exceeds 257 quintillion. A quintillion is 1018  (5 +1)x3 = 18 zeros

    On one cold rainy October afternoon in 1903, Frank Cole was scheduled to give a talk at a meeting of the American Mathematical Society with the unassuming title, "On the Factorization of Large Numbers." When his time came, Cole strode confidently to the blackboard and carefully wrote out:

    267-1 = 147,573,952,589,676,412,927 

    Then he moved to another section of the board and began the long process of multiplying two large numbers together: 193707721 and 761838257287.

    Cole never spoke a word during the tedious multiplication process. But when he finally laid down his chalk and the product agreed with the original decimal expansion of 267-1:

    267-1 = 147573952589676412927   =  193707721 * 761838257287 = 147573952589676412927

    The crowd recognized his accomplishment and broke out in applause. At the meeting, no one asked Cole a single question about his "talk," but later he stated that it had required 20 years of continuous Sunday afternoons to find the factorization of 267 - 1. (It takes my computer with a Pentium 4 processor exactly 125 milliseconds.)

    Frank Nelson Cole's factorization accomplishment is now so well-known that an account of it usually makes an appearance in every modern number theory book written for non-mathematicians.

    A list of Mersenne Primes is at

    357,686,312,646,216,567,629,137 is a prime which when successively "beheaded" is also a prime.  This prime  is over 357 sextillion. A sextillion is 1021  (6 + 1)x3 = 21 zeros

























    2,235,197,406,895,366,368,301,560,000 to 1 are the odds against all four players in bridge being dealt a single suit, over 2.25 octillion to 1. An octillion is 1027     (8 +1)x3 = 27 zeros

    2257 - 1  is a large Mersenne number, but is not a prime.  The Mersenne conjecture, was a statement by Marin Mersenne in his Cogitata Physica-Mathematica (1644; see e.g. Dickson 1919) that the numbers 2^n - 1 were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and were composite for all other positive integers n < 258. (He missed n = 61) Due to the size of these numbers, Mersenne did not and could not test all of them, nor could his peers in the 17th century.  By 1947 Mersenne's range, n < 258, had been completely checked and it was determined that the correct list is  n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127.  In 1903, Frank Cole showed  when n =67, it can be factored into two huge factors.  In 1922, Krait showed when n =257, it can be factored. It has four huge factors. For Mersenne primes, see

    Large Numbers

    NAME Short Scale Long Scale
    Million 106 106
    Milliard   109
    Billion 109 1012
    Trillion 1012 1018
    Quadrillion 1015 1024
    Quintillion 1018 1030
    Sextillion 1021 1036
    Septillion 1024 1042
    Octillion 1027 1048
    Nonillion 1030 1054
    Decillion 1033 1060
    Undecillion 1036 1066
    Duodecillion 1039 1072
    Tredecillion 1042 1078
    Quattuordecillion 1045 1084
    Quindecillion 1048 1090
    Sexdecillion 1051 1096
    Septendecillion 1054 10102
    Octodecillion 1057 10108
    Novendecillion 1060 10114
    Vigintillion 1063 10120
    Centillion 10303 10600
    Googol 10100
    Googolplex 10Googol
    * Short Scale = North America & Modern British
             ** Long Scale = Europe & Older British?

    When a number such as 1045 needs to be referred to in words, it is simply read out: "ten to the forty-fifth". This is just as easy to say, easier to understand, and less ambiguous than "quattuordecillion", which means something different in the long scale and the short scale.

    An easy way to find the value of the above numbers in the short scale (as well as the number of zeroes needed to write them) is to take the number indicated by the prefix (such as 2 in billion, 4 in quadrillion, 18 in octodecillion, etc.), add one to it, and multiply that result by 3. For example, in a trillion, the prefix is tri, meaning 3. Adding 1 to it gives 4. Now multiplying 4 by 3 gives us 12, which is the power to which 10 is to be raised to express a short-scale trillion in scientific notation: one trillion = 1012.

    In the long scale, this is done simply by multiplying the number from the prefix by 6. For example, in a billion, the prefix is bi, meaning 2. Multiplying 2 by 6 gives us 12, which is the power to which 10 is to be raised to express a long-scale billion in scientific notation: one billion = 1012. The intermediate values (billiard, trilliard, etc.) can be converted in a similar fashion, by adding ½ to the number from the prefix and then multiplying by six. For example, in a septilliard, the prefix is sept, meaning 7. Multiplying 7½ by 6 yields 45, and one septilliard equals 1045. Doubling the prefix and adding one then multiplying the result by three would give the same result.







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