

A134307


Primes p such that A^(p1) == 1 (mod p^2) for some A in the range 2 <= A <= p1.


13



11, 29, 37, 43, 59, 71, 79, 97, 103, 109, 113, 127, 131, 137, 151, 163, 181, 191, 197, 199, 211, 223, 229, 233, 241, 257, 263, 269, 281, 283, 293, 307, 313, 331, 347, 349, 353, 359, 367, 373, 379, 397, 401, 419, 421, 433, 439, 449, 461, 463, 487, 499, 509
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OFFSET

1,1


COMMENTS

It's worth observing that there are p1 elements of order dividing p1 modulo p^2 that are of the form r^(k*p) mod p^2 where r is a primitive element modulo p and k=0,1,...,p2. Heuristically, one can expect that at least one of them belongs to the interval [2,p1] with probability about 1  (1  1/p)^(p1) ~= 1  1/e.
Numerically, among the primes below 1000 (out of the total number pi(1000)=168) there are 103 terms of the sequence, and the ratio 103/168 = 0.613 which is already somewhat close to 11/e ~= 0.632.
If we replace p^2 with p^3, heuristically it is likely that the sequence is finite (since 1  (1  1/p^2)^(p1) tends to 0 as p grows).  Max Alekseyev, Jan 09 2009
Replacing p^2 with p^3 gives just the one term (113) for p < 10^6.  Joerg Arndt, Jan 07 2011
If furthermore the number A can be taken to be a primitive root modulo p, i.e., A is a generator of (Z/pZ)*, then that p belongs to A060503.  Jeppe Stig Nielsen, Jul 31 2015


REFERENCES

L. E. Dickson, History of the theory of numbers, vol. 1, p. 105.


LINKS

T. D. Noe, Table of n, a(n) for n=1..1000
W. Keller and J. Richstein Fermat quotients that are divisible by p.


EXAMPLE

Examples (pairs [p, A]):
[11, 3]
[11, 9]
[29, 14]
[37, 18]
[43, 19]
[59, 53]
[71, 11]
[71, 26]
[79, 31]
[97, 53]


MATHEMATICA

Select[ Prime[ Range[100]], Product[ (PowerMod[a, #  1, #^2]  1), {a, 2, #  1}] == 0 &] (* Jonathan Sondow, Feb 11 2013 *)


PROG

(PARI)
{ forprime (p=2, 1000,
for (a=2, p1, p2 = p^2;
if( Mod(a, p2)^(p1) == Mod(1, p2), print1(p, ", ") ; break() );
); ); }


CROSSREFS

Cf. A001220, A055578, A039678, A143548, A222184, A060503.
Sequence in context: A124110 A153768 A092194 * A279775 A211191 A240678
Adjacent sequences: A134304 A134305 A134306 * A134308 A134309 A134310


KEYWORD

nonn


AUTHOR

Joerg Arndt, Aug 27 2008


STATUS

approved



