CH. Il] ARITHMETICAL RECREATIONS 39 



and the corresponding perfect numbers are 6, 28, 496, 8128, 

 33550336, 8589869056, 137438691328, 2305843008139952128, 

 and 2658455991569831744654692615953842176. 



Euler's Biquadrate Theorem*. Another theorem, 

 believed to be true but as yet unproved, is that the arith- 

 metical sum of the fourth powers of three numbers cannot 

 be the fourth power of a number; in other words, we cannot 

 find values of x, y, z, v, which satisfy the equation x* + y* + z* = v*. 

 The proposition is not true of an algebraical sum, for Euler gave 

 more than one solution of the equation x* + y* = z* + v i , for 

 instance, a; =542, y=103, = 359, v = 514. 



Goldbach's Theorem. Another interesting question is 

 whether all even integers can be expressed as a sum of two 

 primes. Asymptotic expressions for the number of ways of 

 expressing even numbers as the sum of two primes have been 

 suggested and are consistent with numerical tests. This suggests 

 that every even number above some limit is expressible in this 

 way; and if so, unless this limit is less than 5000, up to which 

 number the theorem has been empirically established f, it may 

 be accepted as true. 



Lagrange's Theoremj. Another theorem in higher arith- 

 metic which, as far as I know, is still unsolved, is to the effect 

 that every prime of the form 4m, — 1 is the sum of a prime of 

 the form 4n + 1 and of double another prime also of the form 

 4re + 1 ; for example, 23 = 13 + 2 x 5, Lagrange, however, added 

 that it was only by induction that he arrived at the result. 



Fermat's Theorem on Binary Powers. Fermat enriched 

 mathematics with a multitude of new propositions. With one 

 exception all these have been proved or are believed to be 

 true. This exception is his theorem on binary powers, in which 



* See Euler, Commentationes Arithmeticae Collectae, Petrograd, 1849, vol. i, 



pp. 473 476; vol. u, pp. 450 — 456. A somewhat similar unproved theorem is 



that the sum of n numbers of the form x k cannot be of the form u k if n is less 

 than k. 



t Transactions of the Halle Academy (Naturforschung), vol. lxxii, Halle, 

 1897, pp. 6 — 214: see also L'Interme'diaire des Mathimaticiens, 1903, vol. x, 

 and 1904, vol. xi. 



J Nouveaux MSmoires de VAcadimie Boyale des Sciences, Berlin, 1775, p. 356. 



