trigonometry & functions |
analysis | sets & logic | number theory | recreational | misc | nomenclature & history | physics
Not all such pairs of factors are interesting, because...
(2008-01-14) Quadratic formulas giving long sequences of primes.
Quadratic polynomials giving long sequences of prime numbers.
Since P(n-1) = P(-n) = n2 - n + 41 (Legendre, 1798) the above prime values of P(n) are duplicated when n goes down from -1 to -40. So, there are 80 consecutive values of n (from -40 to +39) which make P(n) prime (each such prime number being obtained twice).
Thus, the polynomial n2 - (2q-1) n + (41+q2-q) = (n-q)2 + (n-q) + 41 yields prime values for all integers from 0 to 39+q, provided that q is between 0 and 40. In particular (for q=40) the polynomial n2 - 79n + 1601 yields only prime values as n goes from 0 to 79 (namely, 40 prime values appearing twice each) as observed by Hardy and Wright in 1979.
(2004-04-02) The Area under a Gaussian Curve (Gaussian integral)
A definite integral whose exact value is obtained with a unique method.
The challenge is to compute the integral I = ò e-x2 dx which represents the area under some Gaussian curve. The trick is to consider the square of this integral, which can be interpreted as a 2-dimensional integral which begs to be worked out in polar coordinates... The result involves the constant p.
I 2 =
ò e-x 2 dx
ò e-y 2 dy
e-( x 2 +y 2 ) dx dy
e-r 2 dr
Therefore, the mystery integral I = ò e-t 2 dt is simply equal to Öp
Changing the variable of integration from t to x with t = Öp x yields:
Thus, the function e - p x 2 is a probability distribution whose variance is:
[ HINT: ( - x / 2p ) ( - 2p x e - p x 2 dx ) is easily integrated by parts. ]
Properly scaling the above gives the general expression of a normal Gaussian probability distribution of standard deviation s (and zero mean) :
(2007-04-18) Exceptional simple Lie groups : E6 E7 E8 F4 and G2
5 exceptions to the 4 basic classes of Lie groups : An Bn Cn and Dn
The most complicated is E8 (which may describe fundamental aspects of physical reality). It describes the 248 ways to rotate an object with 57 dimensions.
The so-called Atlas Project culminated in an optimized computation about the representations of E8 which took 77 hours of supercomputer time to complete, on January 8, 2007. The output was a square matrix of order 453060, having entries in a set of 1181642979 distinct polynomials totalizing 13721641221 integer coefficients with values up to 11808808... all packed in 60 GB of data.
(2007-05-07) Monstrous Moonshine (Simon Norton & John Conway)
A 1978 remark about 196884, made by John McKay to John Thompson.
The Fischer-Griess Monster Group is also known as Fischer's Monster, or simply the Monster Group. It's (by far) the largest of the sporadic groups.
It was predicted independently by Bernd Fischer and Robert L. Griess in 1973. Calling it the Friendly Giant, Griess constructed it explicitely in 1981, as the automorphism group of a 196883-dimensional commutative nonassociative algebra over the rational numbers.
In 1978 (before that proof was completed) John McKay (1939-) spotted the appearance of the number 196884 in an expansion of the modular j-function (A000521) and subsequently wondered about some unexpected relation with the Monster, in a letter to John Thompson (himself famous for the 1963 Feit-Thompson Theorem, which paved the road for a 20-year effort resulting in the final classification of finite groups).