# [seqfan] Re: New pseudosquares and pseudocubes found: to which OEIS seq are these additions

Charles Greathouse charles.greathouse at case.edu
Wed Jan 20 05:19:08 CET 2010

```The sequence is A002189, but I'm afraid I added them some time ago.
The reference should be added, though, now that it's "in print".

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Tue, Jan 19, 2010 at 11:11 PM, Jonathan Post <jvospost3 at gmail.com> wrote:
> Just posted on arXiv, dated Jan 19, 2010.
>
> Let (x/y) denote the Legendre symbol [5]. For an odd prime p, let Lp,2, the
> pseudosquare for p, be the smallest positive integer such that
> 1. L_p,2 congruent to 1 (mod 8),
> 2. (L_p,2/q) = 1 for every odd prime q <= p, and
> 3. L_p,2 is not a perfect square.
> In other words, L_p,2 is a square modulo all primes up to p, but is
> not a square.
> We found the following new pseudosquares:
>
> p..................L_p,2
> 367..............36553 34429 47705 74600 46489
> 373..............42350 25223 08059 75035 19329
> 379..............> 10^25
>
> The two pseudosquares listed were found in 2008 in a computation that went up
> to 5 × 1024, taking roughly 3 months wall time. The final computation leading
> to the lower bound of 1025 ran for about 6 months, in two 3-month pieces, the
> second of which finished on January 1st, 2010.
> Wooding and Williams [11] had found a lower bound of L_367,2 > 120120 ×
> 2^64 ~ 2.216 × 102^4. (Note: a complete table of pseudosquares,
> current as of this
> writing, is available at http://cr.yp.to/focus.html care of Dan Bernstein).
> Similarly, for an odd prime p, let Lp,3, the pseudocube for p, be the
> smallest...
>
> http://arxiv.org/abs/1001.3316
>    Title: Sieving for pseudosquares and pseudocubes in parallel using
> doubly-focused enumeration and wheel datastructures
>    Authors: Jonathan P. Sorenson
>    Subjects: Number Theory (math.NT)
>
>    We extend the known tables of pseudosquares and pseudocubes,
> discuss the implications of these new data on the conjectured
> distribution of pseudosquares and pseudocubes, and present the details
> of the algorithm used to do this work. Our algorithm is based on the
> space-saving wheel data structure combined with doubly-focused
> enumeration, run in parallel on a cluster supercomputer.
>
>
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