Some results on pseudosquares

Abstract

Abstract. If p is an odd prime, the pseudosquare Lp is defined to be the least positive nonsquare integer such that Lp ≡ 1 (mod 8) and the Legendre symbol (Lp/q) = 1 for all odd primes q ≤ p. In this paper we first discuss the connection between pseudosquares and primality testing. We then describe a new numerical sieving device which was used to extend the table of known pseudosquares up to L 271 . We also present several numerical results concerning the growth rate of the pseudosquares, results which so far confirm that Lp > e √ p/2 , an inequality that must hold under the extended Riemann Hypothesis.

Doubly focused enumeration of locally square polynomial values

“…If x (r−1)/2 mod r = r − 1 then x is locally non-square at r. If x (r−1)/2 mod r = 0 then x is divisible by r but, being squarefree, not by r 2 , so it is locally non-square at r. A series of previous computations, initiated by Kraitchik in 1924 and continued by Lehmer, Lehmer, Shanks, Patterson, Williams, Stephens, and Lukes, showed with considerably more effort that r ≤ 281 for all x up to about 7 · 10 19 ≈ 2 66 . See [9], [10], [11], [17], [14, page 134], and [15]. For example, the computation of Lukes, Patterson, and Williams in [15] was a focused enumeration of all small y such that 1 + 24y is a non-unit square modulo m 1 = 5 · 7 · 11 · 13; there are about H/27 such values of y in [1, H].…”

“…See [9], [10], [11], [17], [14, page 134], and [15]. For example, the computation of Lukes, Patterson, and Williams in [15] was a focused enumeration of all small y such that 1 + 24y is a non-unit square modulo m 1 = 5 · 7 · 11 · 13; there are about H/27 such values of y in [1, H]. My computation was a doubly focused enumeration, as explained in Section 2 and Section 3, of all small y such that 1 + 24y is a non-unit square modulo both 25 .…”

“…Lehmer in [10] defined, for each prime q ≥ 3, the corresponding "pseudo-square" L q as the smallest positive nonsquare integer x ∈ 1 + 8Z for which r > q, i.e., for which x (p−1)/2 mod p = 1 for all primes p ≤ q. The same terminology is used in [18, page 522], [14], and [15].…”

The least inert prime in a real quadratic field

“…In [6], Granville, Mollin and Williams prove the following theorem: For the computational aspect, they used the Manitoba Scalable Sieving Unit, a very powerful sieving machine (see [8] for more details). They ran the machine for a period of 5 months to produce three tables.…”

The Pseudosquares Prime Sieve

“…It uses an Eratosthenes-like sieve followed by the pseudosquares prime test of Lukes, Patterson, and Williams [14] (which effectively includes a base-2 pseudoprime test). Our sieve has a conjectured running time of O(n log n) arithmetic operations and O((log n)…”