D. J. Bernstein
More number-theoretic computations

Doubly focused enumeration of locally square polynomial values

Papers

[focus] 8pp. (PDF) (PS) (DVI) D. J. Bernstein. Doubly focused enumeration of locally square polynomial values. Document ID: b4795a4f12863c26de5b7afe9296ffd8. URL: http://cr.yp.to/papers.html#focus. Date: 2003.09.28. Supersedes: (PDF) (PS) (DVI) 2001.12.31. (PDF) (PS) (DVI) 2003.02.03.

Coming soon: focus2d.

Relevant talks: 2003.05.26 (slides available), ``Doubly focused enumeration of locally square polynomial values.'' 2004.06.24 (transcript and slides available), ``Doubly focused enumeration in two dimensions.''

Software

focus, not yet released, is a collection of tools for doubly focused enumeration of locally square values of polynomials.

Locally square integers

Let x be a positive non-square integer with x mod 8 = 1. What is the smallest odd prime p such that x^((p-1)/2) mod p is not equal to 1?

Some bounds:
p is at mostif x is smaller than, but not if x is equal tocredit
3731924 Kraitchik
52411924 Kraitchik
710091924 Kraitchik
1126411924 Kraitchik
1380891924 Kraitchik
17180011924 Kraitchik
19538811924 Kraitchik
23874811924 Kraitchik
291170491924 Kraitchik
315157611924 Kraitchik
3710832891924 Kraitchik
4132066411924 Kraitchik
4338189291924 Kraitchik
4792573291924 Kraitchik
53220008011928 Lehmer
59484738811928 Lehmer
61484738811928 Lehmer
671752442811954 Lehmer
714277333291954 Lehmer
734277333291954 Lehmer
798987162891954 Lehmer
8328055446811970 Lehmer Lehmer Shanks
8928055446811970 Lehmer Lehmer Shanks
9728055446811970 Lehmer Lehmer Shanks
101103102634411970 Lehmer Lehmer Shanks
103236163314891970 Lehmer Lehmer Shanks
107851576104091970 Lehmer Lehmer Shanks
109851576104091970 Lehmer Lehmer Shanks
1131962650950091970 Lehmer Lehmer Shanks
1271962650950091970 Lehmer Lehmer Shanks
1312871842842801Lehmer?
1372871842842801Lehmer?
1392871842842801Lehmer?
149262508870237291973 Lehmer?
151262508870237291973 Lehmer?
1571124347329019691988 Patterson Williams
1631124347329019691988 Patterson Williams
1671124347329019691988 Patterson Williams
1731789362225370811988 Patterson Williams
1791789362225370811988 Patterson Williams
1816961611102090491988 Patterson Williams
1916961611102090491988 Patterson Williams
19328549096481038811989 Stephens Williams
19764500455166307691989 Stephens Williams
19964500455166307691989 Stephens Williams
211116413992479479211989 Stephens Williams
223116413992479479211989 Stephens Williams
2271906214289051864491991 Lukes Williams
2291966402481219286011991 Lukes Williams
2337126243350950935211994 Lukes Patterson Williams
23917738557918778503211994 Lukes Patterson Williams
24123276870641244744411994 Lukes Patterson Williams
25163849918730598366891994 Lukes Patterson Williams
25780192046613054197611994 Lukes Patterson Williams
263101981005820462876891994 Lukes Patterson Williams
269101981005820462876891994 Lukes Patterson Williams
271101981005820462876891994 Lukes Patterson Williams
277698482883209001869691996? Lukes Patterson Williams
2812089363657990449759612001 Bernstein
2835335526633398282036812003? Williams Wooding
2939366640792667146970892003 Williams Wooding
3079366640792667146970892003 Williams Wooding
31121422028603702699161292003 Williams Wooding
31321422028603702699161292003 Williams Wooding
31721422028603702699161292003 Williams Wooding
331136491544915582988032812003 Williams Wooding
337345948588016701277788012003 Williams Wooding
347994929459304792133340492003 Williams Wooding
349994929459304792133340492003 Williams Wooding
3532953634874009003108804012003 Williams Wooding
3592953634874009003108804012003 Williams Wooding
36736553344294770574600464892008 Sorenson
37342350252230805975035193292008 Sorenson
The x bounds in this table are often called ``pseudosquares'' or ``minimal pseudosquares.''