SERIES—CUBES AND OTHERS 37

ference equals [g X9 x10)+1]. This gives a formula which
applies to any pair of cubes % and »3, where x=y-+1. Thus

x¥3—yd=29xy+1
This can be proved by substituting (y+1) for x. Then

(A1 e L gt ey 1000
HIPHYX
=3)(y+1)+1
=3)(¥) +1
=3%)+1
It has already been shown that it is possible to find (in
various different ways) a square number such that it is
equivalent to the sum of two other square numbers. That is,
x2=y2+22. No similar number of a higher power has ever
been found to satisfy the same conditions. Thus, no satis-
factory solution exists for ™ =»" 42" where the exponent » is
greater than 2. The renowned mathematician Fermat claimed
to have found a proof why this should be impossible but his
claim has never been substantiated. Other mathematicians
have proved it to be impossible for many values of #» but no
one has succeeded in producing a general proof for all values
of n.
It is, however, possible to find a cube number equal to the
sum of three other cubes.
3 3 _+_ 43 -+ 5 3_—-(3
13 +63 +83 :93

i
4

R R e

jiltkices

R
TR L

RanHRRTEABanan

T 0 B S RS

X

B A a3 R R T AR A s s
RN 5 n1itiR Bt B REOO !

AR R
R

]