880 
tigator. On the contrary, one of the chief 
differences between the great mathema- 
tician and the poor one is that the former 
ean direct his students into fields which are 
likely to become well known in the near 
future, while the latter can only direct 
them to the well-known standing problems 
of the past, whose approaches have been 
tramped down solid by the feet of the 
mediocre, who are often even too stupid 
to realize their limitations. The best stu- 
dents can work their way through this hard 
crust, but the paddle of the weaker ones 
will only serve to increase its thickness if 
it happens to make any impression what- 
ever. 
It would not be difficult to furnish a long 
list of standing mathematical problems 
of more or less historic interest. Probably 
all would agree that the most popular one 
at the present time is Fermat’s greater 
theorem. In fact, this theorem has become 
so popular that it takes courage to mention 
it before a strictly mathematical audience, 
but it does not appear to be out of place 
before a more general audience like this. 
The ancient Egyptians knew that 3*-+ 
4° — 5? and the Hindus Imew several other 
such triplets of integers at least as early as 
the fourth century before the christian era.” 
These triplets constitute positive integra! 
solutions of the equation 
a? a 2 
Pythagoras gave a general rule by means 
of which one can find any desired number 
-of such solutions, and hence these triplets 
are often called Pythagorean numbers. 
Another such rule was given by Plato, 
while Kuclid and Diophantus generalized 
and extended these rules. 
Fermat, a noted French mathematician 
of the seventeenth century, wrote on the 
*Lietzmann, ‘‘Der Pythagoreische Lehrsatz,’’ 
1912, p. 52. 
SCIENCE 
[N.S. Vou. XXXV. No. 910 
margin of a page of his copy of Diophantus 
the theorem that it is impossible to find any 
positive integral solution of the equation 
(n> 2). 
He added that he had discovered a won- 
derful proof of this theorem, but that the 
margin of the page did not afford enough 
room to add it. This theorem has since 
become known as F'ermat’s greater theorem 
and has a most interesting and important 
history, which we proceed to sketch. 
About a century after Fermat had noted 
this theorem Euler (1707-1783) proved it 
for all the cases when m is a multiple of 
either 3 or 4, and, during the following 
century, Dirichlet (1805-1859) and Le- 
gendre (1752-1833) proved it for all the 
eases when 7 is a multiple of 5. The most 
important step towards a general proof was 
taken by Kummer (1810-1893), who ap- 
plied to this problem the modern theory of 
algebraic numbers and was thus able to 
prove its truth for all multiples of primes 
which do not exceed 100 and also for all 
the multiples of many larger primes. 
The fact that such eminent mathemati- 
cians as Fermat, Euler, Dirichlet, Legendre 
and Kummer were greatly interested in 
this problem was sufficient to secure for it 
considerable prominence in mathematical 
literature, and several mathematicians, in- 
eluding Dickson, of Chicago, succeeded in 
extending materially some of the results 
indicated above. The circle of those taking 
an active interest in the problem was sud- 
denly greatly enlarged, a few years ago, 
when it become known that a prize of 
100,000 Marks (about $25,000) was await- 
ing the one who could present the first com- 
plete solution. This amount was put in 
trust of the Gottingen Gesellschaft der 
gn +f yr = gn 
*Fermat’s words are as follows: ‘‘Cujus re1 
demonstrationem mirabilem sane detexi. Hane 
Imarginis exiguitas non caperet.’’ 
