Modular Invariants.-For example. consider where p is a prime. If x, y, z are linear functions of X, Y, Z with integral coefficients whose determinant is d, then l-dL is divisible by p. We therefore call / a modular invariant. But 1 has the factor x. It follows that 1 differs by a multiple of p from the product of all linear functions of x, y, z whose coefficients are chosen from o, 1, , p- I such that the first coefficient >o is I. These facts hold also when 1 is replaced by the like deter minant of order n. For n= 2, the last theorem implies Fermat's theorem.
All algebraic invariants (see ALGEBRAIC FORMS) with integral coefficients are modular invariants ; but the converse is not true. The simplest theory of modular invariants is based on the modular classes of forms. For an exposition with bibliography, see Dickson's Invariants in the Theory of Numbers (American Mathematical Society Colloquium Lectures, 1914).
The number of primes is infinite (Euclid, Elem., ix. 2o). Any arithmetical progression mx+n contains an infinitude of primes if m and n are relatively prime. The first proof is due to Dirichlet (1837), who extended the theorem to numbers a+bi such that a and b are integers. He proved that q=
represents infinitely many primes if a, 2b, c have no common factor. Also an infinitude are rep resented simultaneously by f and mx+n provided the two can represent the same number and m and n are relatively prime.
E> 0 and any relatively prime integers m and n, the number of primes mx+n which lie between z and (I+ )z increases to infinity with z. Take E= I. Hence if the integer y exceeds a certain limit (6, in fact), there exists at least one prime p such that ly< p < y 2 ; this is known as Bertrand's postulate (1845) In 1742 Goldbach conjectured that every positive even integer is a sum of two primes. This has been verified to io,000, but not yet proved. Subject to assumptions about the roots of Dirichlet's function L, Hardy and Littlewood recently proved that all sufficiently large even integers e are sums of three odd primes; also if N of the first m even positive integers are not sums of two odd primes, N/m approaches zero as m increases indefi nitely. By developing their first proof, Lucke showed that every e> (3.6)1o" is a sum of three odd primes.
x is sufficiently large, the number ir(x) of primes < x is between 092129Q and i Io555Q, where Q = x/logx. By use of Riemann's function c(s)
Hadamard and de la Vallee-Poussin proved independently in 1896 that the sum of the natural logarithms of all primes x is equal to x asymptotically. This implies the important result that ir(x) is asymptotic to Q. If we integrate du/logu first from o to I -.5 and second from I-FS to x, add the results, and take the limit of the sum for 6= o, we get the integral logarithm of x, denoted by Li(x). It is asymptotic to Q and hence to ir(x). But Li(x) represents r(x) more exactly than does Q or Q+x/logx, etc.
Hand buch der Lehre von der V erteilung der Primzahlen (1909).
There are just two partitions I + 5 and 2 + 4 of 6 into two parts chosen from 1, 2, 3, without repetition; there is the third partition 3+3 if repetition is allowed. If we exhibit the partition 2+4 as 11 and then sum the columns
instead of the lines, we get the partition 2+2+1+1. If we repeat the process on the latter, we return to 4 + 2. This proves that the number of partitions of n into m parts is equal to the number of partitions of n in which m is the largest part. Taking m 1, , k, we conclude that the number of partitions of n into not more than k parts is equal to the number of partitions of n with no part greater than k.
The number of partitions of n into m parts chosen without repetition from the distinct positive integers , Cr is the
coefficient of x" zm in the expansion of the product (I
(I +xcrz). Next, the coefficient of x" zm in the series giving the expansion of (1
(1
is the number of partitions of n into m parts chosen from
, c,., repetitions allowed.
D= (1 -x) (i -
(i -xm), the coefficient of x" in 1/D is the number (n, m) of partitions of n into parts
m not necessarily distinct. The relation (n, m) = (n, m - I) + (n - m, m) serves to compute (n, m).
There are important applications of Sylvester's theorem (1855) concerning the number Q of partitions of n into distinct positive integers
, Cr repetitions allowed, whence Q is the number of sets of integral solutions o of
Then
where
is the coefficient of r/t in the development in ascending powers of t of in which the summation extends over the various primitive qth roots p of unity. Thus
= o unless q divides some ci.