A355962 -id:A355962

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A355959

Primes p such that (p+2)^(p-1) == 1 (mod p^2).

+10
7

5, 45827

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(3) > 107659373057 if it exists.

Primes p such that the Fermat quotient of p base 2 (A007663) is congruent to 1/2 modulo p. - Max Alekseyev, Aug 27 2023

LINKS

Table of n, a(n) for n=1..2.

PROG

(PARI) forprime(p=1, , if(Mod(p+2, p^2)^(p-1)==1, print1(p, ", ")))

CROSSREFS

(p+k)^(p-1) == 1 (mod p^2): A355960 (k=5), A355961 (k=6), A355962 (k=7), A355963 (k=8), A355964 (k=9), A355965 (k=10).

Cf. A007663.

KEYWORD

nonn,hard,more,bref

AUTHOR

Felix Fröhlich, Jul 21 2022

STATUS

approved

A355960

Primes p such that (p+5)^(p-1) == 1 (mod p^2).

+10
6

3, 23, 1574773

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Equivalently, primes p such that 5^p == p+5 (mod p^2), or Fermat quotient q_p(5) == 1/5 (mod p). - Max Alekseyev, Sep 16 2024

LINKS

Table of n, a(n) for n=1..3.

PROG

(PARI) forprime(p=1, , if(Mod(p+5, p^2)^(p-1)==1, print1(p, ", ")))

CROSSREFS

(p+k)^(p-1) == 1 (mod p^2): A355959 (k=2), A355961 (k=6), A355962 (k=7), A355963 (k=8), A355964 (k=9), A355965 (k=10).

KEYWORD

nonn,hard,more,bref

AUTHOR

Felix Fröhlich, Jul 21 2022

STATUS

approved

A355961

Primes p such that (p+6)^(p-1) == 1 (mod p^2).

+10
6

13, 47, 3803, 151051, 240727, 259933847

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..6.

PROG

(PARI) forprime(p=1, , if(Mod(p+6, p^2)^(p-1)==1, print1(p, ", ")))

CROSSREFS

(p+k)^(p-1) == 1 (mod p^2): A355959 (k=2), A355960 (k=5), A355962 (k=7), A355963 (k=8), A355964 (k=9), A355965 (k=10).

KEYWORD

nonn,hard,more

AUTHOR

Felix Fröhlich, Jul 21 2022

STATUS

approved

A355963

Primes p such that (p+8)^(p-1) == 1 (mod p^2).

+10
6

1531, 7445287

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(3) > 34294200797 if it exists.

LINKS

Table of n, a(n) for n=1..2.

PROG

(PARI) forprime(p=1, , if(Mod(p+8, p^2)^(p-1)==1, print1(p, ", ")))

CROSSREFS

(p+k)^(p-1) == 1 (mod p^2): A355959 (k=2), A355960 (k=5), A355961 (k=6), A355962 (k=7), A355964 (k=9), A355965 (k=10).

KEYWORD

nonn,hard,more,bref

AUTHOR

Felix Fröhlich, Jul 21 2022

STATUS

approved

A355964

Primes p such that (p+9)^(p-1) == 1 (mod p^2).

+10
6

13, 19, 2207, 26041, 332698495781

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(6) > 10^13 if it exists. - Jason Yuen, May 12 2024

LINKS

Table of n, a(n) for n=1..5.

PROG

(PARI) forprime(p=1, , if(Mod(p+9, p^2)^(p-1)==1, print1(p, ", ")))

CROSSREFS

(p+k)^(p-1) == 1 (mod p^2): A355959 (k=2), A355960 (k=5), A355961 (k=6), A355962 (k=7), A355963 (k=8), A355965 (k=10).

KEYWORD

nonn,hard,more

AUTHOR

Felix Fröhlich, Jul 21 2022

EXTENSIONS

a(5) from Jason Yuen, May 12 2024

STATUS

approved

A355965

Primes p such that (p+10)^(p-1) == 1 (mod p^2).

+10
6

13, 41, 97, 809, 1151, 1657, 800011

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

A computer search taking less than 3 seconds shows there are no further terms below the one millionth prime. - Harvey P. Dale, Mar 04 2024

I ran the PARI program below for 8.5 hours and it did not find any further terms. (I do not know how far it searched.) - N. J. A. Sloane, Mar 05 2024

LINKS

Table of n, a(n) for n=1..7.

MATHEMATICA

Select[Prime[Range[70000]], PowerMod[#+10, #-1, #^2]==1&] (* Harvey P. Dale, Mar 04 2024 *)

PROG

(PARI) forprime(p=1, , if(Mod(p+10, p^2)^(p-1)==1, print1(p, ", ")))

CROSSREFS

(p+k)^(p-1) == 1 (mod p^2): A355959 (k=2), A355960 (k=5), A355961 (k=6), A355962 (k=7), A355963 (k=8), A355964 (k=9).

KEYWORD

nonn,hard,more

AUTHOR

Felix Fröhlich, Jul 21 2022

STATUS

approved

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