Beauty in Mathematics

La rivista The Mathematical Intelligencer nel n. 4/1988 propose ai suoi lettori la seguente lista di 24 risultati al fine di effettuare una indagine sulle formule ritenute più belle.

 A Euler’s formula for a polyhedron: V + F =E + 2. M At any party, there is a pair of people who have the same number of friends present. B Any square matrix satisfies its characteristic equation N The number of partitions of an integer into odd integers is equal to the number of partitions into distinct integers. C 5{(1 – x5)(1 – x10)(1 – X15) . . . }5  {(1 – x)(1 – x2)(1 – x3 )(1 – x4). . . }6           = p(4) + p(9)x+ p(14)x2 +. . . where p(n) is the number of partitions of n. O If the points of the plane are each coloured red, yellow, or blue, there is a pair of points of the same colour of mutual distance unity. D The number of primes is infinite. P Every plane map can be coloured with 4 colours. E There is no rational number whose square is 2 Q A continuous mapping of the closed unit disk into itself has a fixed point. F Every prime of the form 4n + 1 is the sum of two integral squares in exactly one way. R Write down the multiples of root 2, ignoring fractional parts, and underneath the numbers missing from the first sequence. 1    2    4    5     7     8     9      11     12 … 3    6   10  13  17   20    23     27     30 … The difference is 2n in the nth place. G 1 + 1/22 + 1/32 + 1/42 + 1/52 + … =  p 2/6. S A regular icosahedron inscribed in a regular octagon divides the edges in the Colden Ratio. H 1         –         1        +         1 2 x 3 x 4        4 x 5 x 6         6 x 7 x 8– ….. = p – 3 .  4 T The number of representations of an odd number as the sum of 4 squares is 8 times the sum of its divisors; of an even number, 24 times the sum of its odd divisors. I p  is transcendental. U The word problem for groups is unsolvable. J Every number greater than 77 is the sum of integers, the sum of whose reciprocals is l. V The order of a subgroup divides the order of the group. K The maximum area of a quadrilateral withsides a, b, c, d is {(s-a)(s-b)(s-c)(s-d)}½,where s is half the perimeter. W eip = – 1. L There is no equilateral triangle whose vertices are plane lattice points. X There are 5 regular polyhedra.