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

Jonathan Post jvospost3 at gmail.com
Wed Jan 20 08:14:20 CET 2010

```Charles Greathouse should never apologize to me.  He's done
extraordinary, outstanding work, and I expect to coauthor a journal
paper with him, on Decimal Goedelization. As in A100200  Decimal
Goedelization of antitheorems from propositional calculus, in Richard
Schroeppel's metatheory of A101273; and related seqs.

In this case, numbers do appear, and only later formally in print, so
it would be appropriate to add that citation.

Thank you again!

-- Prof. Jonathan Vos Post

On Tue, Jan 19, 2010 at 8:19 PM, Charles Greathouse
<charles.greathouse at case.edu> wrote:
> 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.
>>
>>
>> _______________________________________________

>

```