196 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICSthat complex numbers of the form an + á\è + • • • + áç-éèç~é, where èç = 1 and
áõ,.. -, a„_i are integers, can be factored uniquely, just like ordinary integers. Ernst
Eduard Kummer (1810-1893) had noticed some 10 years earlier that such is not
the case. This was just one of the many ways in which the objects studied by
mathematicians became increasingly abstract, and the old objects of numbers and
space became merely special cases of the general objects about which theorems are
proved. Kummer was the first to make general progress toward a proof of Fermat's
last theorem. The conjecture that xp + yp — zp has no solutions in positive integers
x, y, and æ when ñ is an odd prime had been proved only for the cases ñ = 3, 5,
and 7 until Kummer showed that it was true for a class of primes called regular
primes, which included all the primes less than 100 except 37, 59, and 67. This
step effectively closed off the possibility that Fermat might be proved wrong by
calculating a counterexample.
1.9. The prime number theorem. A good estimate of the number of primes less
than or equal to a given integer Í is given by N/(\og TV). This estimate follows from
the unproved estimate of Gauss given above. The estimate suggested by Legendre,
N/(AlogN + B) with A = 1, Β = -1.08366, turns out to be correct only in
its first term. This fact was realized by Dirichlet, but only after he had written
approvingly of the estimate in print. (He corrected himself in a marginal note
on a copy of his paper given to Gauss.) Dirichlet suggested X^=2 (V0°S^)) 85 a
better approximation. This problem was also studied by the Russian mathematician
Pafnutii L'vovich Chebyshev (1821-1894).^8 In 1851 Chebyshev proved that if
a > 0 is any positive number (no matter how small) and m is any positive number
(no matter how large), the inequality
. fn dx an
ð(ç)>
/ 2 ^Ã÷-ß^-ç
holds for infinitely many positive integers n, as does the inequality
.. fn dx an
J 2 log ÷ log ç
This result suggests that ð(ç) ~ [n/(lnn)], but it would be desirable to know if
there is a constant A such that
/ í An
7r(n = + åç,
logn
where åç is of smaller order than ð(ç). It would also be good to know the rate at
which åç/ð(ç) tends to zero. Chebyshev's estimates imply that if A exists, it must
be equal to 1, and as a result, Legendre's approximation cannot be valid beyond
the first term. Chebyshev was able to show that
0.92129 < < 1.10555.
lognChebyshev mentions only Legendre in his memoir on this subject and shows that
his estimates refute Legendre's conjecture. He makes no mention of Gauss, whose
integral Li (x) appears in his argument. Similarly, Riemann makes no mention of
(^8) In Russian this name is pronounced "Cheb-wee-SHAWF," approximately. However, because
he wrote so often in French, where he signed his name as "Tchebycheff," it is usually prounced
"CHEB-ee-shev" in the West.