Emma Lehmer

Connection between Gaussian periods and cyclic units

This paper finds that all known parametric families of units in real quadratic, cubic, quartic and sextic fields with prime conductor are linear combinations of Gaussian periods and exhibits these combinations. This approach is used to find new units in the real quintic field for prime conductors p n4 + 5n3 + 15n2 + 25n + 25.

Integer sequences having prescribed quadratic character

For the odd primes p, = 3, p2 = 5, ■ • -, we determine integer sequences Np such that the Legendre symbol (N/p,) = +1 for all pi S P for a prescribed array of signs ± 1; (i.e., for a prescribed quadratic character). We examine six quadratic characters having special interest and applications. We present tables of these Np and examine some applications, particularly to questions concerning extreme values for the smallest primitive root (of a prime N), the class number of the quadratic field R(J — N), the real Dirichlet L functions, and quadratic character sums.

On runs of residues

According to a theorem of Alfred Brauer [1] all sufficiently large primes have runs of I consecutive integers that are kth power residues, wher-e k and I are arbitrarily given integers. In this paper we consider the question of the first appearance of such runs. Let p be a sufficiently large prime and let r = r(k, 1, p) be the least positive integer such that (1) r, r+ 1, r+ 2,*, r+l-1 are all congruent modulo p to kth powers of integers > 0. It is natural to ask, when k and I are given, how large is this minimum r and are there primes p for which r is arbitrarily large? If we let A(k, 1) = lim sup r(k, 1, p) p 00 c

Pairs of consecutive power residues

Introduction. Until recently none of the numerous papers on the distribution of quadratic and higher power residues was concerned with questions of the following sort: Let k and m be positive integers. According to a theorem of Brauer (1), for every sufficiently large prime p there exist m consecutive positive integers r, r + l , . . . , r + m — 1, each of which is a &th power residue of p. Let r(k, m, p) denote the least such r. What can be said about the behaviour of r(k, m, p) as p varies? In particular, for what values of k and m is r(k, m, p) bounded, and for these values what is its maximum? For a fixed k and m we call a prime exceptional and denote it by p* if there do not exist m consecutive integers each of which is a &th power residue of p*. Set A(k,m) = Max r (k, m, p), where the maximum is taken over all nonexceptional primes p. In a previous paper (4) Lehmer and Lehmer showed that

A new factorization technique using quadratic forms

The paper presents a practical method for factoring an arbitrary N by represent- ing N or XN by one of at most three quadratic forms: XN = x- - Dy2, X = 1,-1, 2, D = -1, ?2, ?3, ?6. These three forms appropriate to N, together with inequalities for y, are given for all N prime to 6. Presently available sieving facilities make the method quite effective and economical for numbers N having 20 to 25 digits. Four examples arising from aliquot series are discussed in detail. It is the purpose of this paper to present and illustrate a new procedure for factor- ing numbers N of no special form which in the present state of the art is particularly effective for numbers having from 20 to 25 decimal digits. The implementation of the method was the result of three circumstances: (a) a decision to assist Richard Guy and John Selfridge in their survey of aliquot series, i.e., sequences of iterates of the sum of the proper divisors of a number, a source of many numbers N of the above- mentioned magnitude; (b) the elimination of idle time at the Computer Center of the University of California, Berkeley campus, which made direct search for the factors of N prohibitively expensive; and (c) the availability of the Delay Line Sieve, DLS-157 at no cost (5). It is hoped that those readers who have unlimited access to the virtuosity of an expensive computer system may also find the method of some use although circumstances (a), (b) and (c) may all fail to exist, since the sieving part of the procedure can easily be done inside the system (6), even if at a slower rate than the million per second rate of the off line DLS-157. The present method is a modification of much older ones depending on the representation of N, or a chosen multiple of N, by a binary quadratic form. Such methods began with Fermat, 1643, (3) who suggested solving

Rational Reciprocity Laws

It is well known that the famous Legendre law of quadratic reciprocity, of which over 150 proofs are in print, has been generalized over the years to algebraic fields by a number of famous mathematicians from Gauss to Artin to the extent that it has become virtually unrecognizable. On the other hand, it seems to have escaped notice that in the past decade there were developed rational reciprocity laws for higher power residues which are more direct and easily recognizable generalizations of the Legendre law. These recent developments will be the subject of this report.

Cyclotomy with short periods

This paper develops cyclotomy for periods of lengths 2, 3 and 4 for moduli which are primes and products of two primes.

Period equations applied to difference sets

has exactly X solutions for d 0 (mod v). A multiplier of a difference set is any number t such that the set tal, ta2, * , tak is congruent to the set al+s, a2+s, , * ak+s in some order, for some value of s. Hall and Ryser[1] proved the following interesting theorem: Every prime divisor q of k -X is a multiplier provided q >X. Although the proviso q>X is essential to the proof of the theorem, it appears that all divisors of k-X are actually multipliers in all the explicit numerical examples of difference sets which are available. It therefore seems of interest to test this theorem out more generally on classes of known residue difference sets, that is, difference sets composed of nth power residues modulo a prime p. In this case we have shown [2] that: The set of multipliers of a residue difference set is the set itself. Therefore any statement we can make about multipliers of a residue difference set will also be valid for the residues themselves and vice versa. It is well known that the (p 1)/2 = k quadratic residues modulo a prime p -1 (mod 4) form a difference set of multiplicity = (p-3)/4, so that k-X=(p+1)/4. The validity of Hall and Rysers theorem for all the divisors of k -X follows for these sets from the rather trivial theorem to the effect that:

On a resultant connected with Fermat's last theorem

In his book on Fermats Last Theorem Bachmannf proved that if p is an odd prime and if Ap_i is not divisible by p z } then Fermats equation x+y+z = 0 has no solution (x, y, z) prime to p. S. LubelskyJ proved in a recent paper, using the distribution of quadratic residues, that if p^7, Ap_i is not only divisible by py but by p , thus annulling Bachmanns criterion except for p = 3 and p = 5. We shall now show how, by a straightforward manipulation with the above determinant, one can prove much more.

A family of supplementary difference sets

This note exhibits a family of m – ( ν , k , λ) supplementary difference sets with parameters ν = 2 mf + 1, where ν = p is a prime and f is odd, k = mf and λ = m ( mf −1)/2. These sets are composed of a union of cosets of 2 m -th power residues of p .