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A new solution for the equation tau(p)=3D0 mod p.
After p=3D2,3,5,7 and 2411 (Newman) we just discover (03/15/2010)
the following value p =3D 7758337633

Date: Mon, 15 Mar 2010 23:18:19 -0500
Reply-To: Number Theory List <[log in to unmask]>,
Lygeros <[log in to unmask]>
Sender: Number Theory List <[log in to unmask]>
From: Lygeros <[log in to unmask]>
Subject: A new solution for the equation tau(p)=0 mod p.
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A new solution for the equation tau(p)=3D0 mod p.

After p=3D2,3,5,7 and 2411 (Newman) we just discover (03/15/2010)
the following value p =3D 7758337633

tau(p) =3D 3634118031125820057253378550628821747860472052772622882=20
=3D 2 * 31481 * 7758337633 * 7439638579196209777834920016764711229=
817
=3D 0 mod p

Verification of congruences :

p =3D 1 mod 8
tau(p) =3D 546 mod 2^11
sigma(p,11) =3D 546 mod 2^11

p =3D 1 mod 3
tau(p) =3D 185 mod 3^6
sigma(p,1231)*p^-610 =3D 185 mod 3^6

p =3D 3 mod 5
tau(p) =3D 7 mod 5^3
sigma(p,71)/(p^30) =3D 7 mod 5^3

p =3D 4 mod 7
tau(p) =3D 1 mod 7
p*sigma(p,9) =3D 1 mod 7

tau(p) =3D 48 mod 691
sigma(p,11) =3D 48 mod 691

We dedicate our result to Jean-Pierre Serre.

N. Lygeros and O. Rozier

http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1003&L=nmbrthry&T=0&F=&S=&X=4FFB4C1EA1A27AB190&Y=nlygeros@gmail.com&P=680