A354155 Lagrange primes: primes p == 1 (mod 4) such that X = (p-1)/2 is the least solution in the interval [1, (p-1)/2] of the congruence (X!)^2 == -1 (mod p).

5, 13, 17, 29, 41, 53, 73, 97, 137, 229, 241, 257, 281, 313, 397, 409, 449, 457, 461, 521, 541, 617, 653, 661, 677, 733, 769, 809, 853, 857, 929, 953, 997, 1021, 1069, 1109, 1201, 1213, 1217, 1249, 1277, 1361, 1373, 1409, 1489, 1553, 1597, 1609, 1621, 1697

Offset

1,1

References

J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.5.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..3100

Prog

(Python)
from itertools import islice
from sympy import factorial, nextprime
def agen(): # generator of terms
    p = 5
    while True:
        X = (p-1)//2
        Xf = factorial(X)**2
        if all(pow(factorial(Y), 2, p)+1 != p for Y in range(X-1, 0, -1)):
            yield p
        p = nextprime(p)
        while p%4 != 1:
            p = nextprime(p)
print(list(islice(agen(), 5))) # Michael S. Branicky, May 30 2022
(PARI) is(n)=if(n%4 != 1 || !isprime(n), return(0)); my(t1=lift(sqrt(Mod(-1, n))), t2=n-t1, t=Mod(1, n)); for(k=2, n\2, if(t==t1 || t==t2, return(0)); t*=k); 1 \\ Charles R Greathouse IV, Aug 03 2023
(PARI) list(lim)=my(v=List()); forprimestep(p=5, lim\1, 4, my(t1=lift(sqrt(Mod(-1, p))), t2=p-t1, t=Mod(1, p)); for(k=2, p\2, if(t==t1 || t==t2, next(2)); t*=k); listput(v, p)); Vec(v) \\ Charles R Greathouse IV, Aug 03 2023

Crossrefs

Cf. A002144, A354156.

Sequence in context: A307096 A283391 A145016 * A123079 A273950 A268511

Adjacent sequences: A354152 A354153 A354154 * A354156 A354157 A354158

Keyword

nonn

Author

N. J. A. Sloane, May 29 2022, based on Section 18.5 of Cosgrave (2022)

Extensions

a(20) and beyond from Michael S. Branicky, May 30 2022

Status

approved