A254678 Primes p with the property that there are four consecutive integers less than p whose product is 1 mod p.

7, 17, 23, 31, 41, 47, 73, 89, 97, 103, 127, 137, 151, 167, 199, 223, 233, 239, 241, 257, 271, 281, 311, 313, 353, 359, 367, 383, 409, 431, 433, 439, 449, 479, 487, 503, 521, 577, 593, 601, 607, 647, 673, 719, 727, 743, 751, 761, 769, 839, 857, 881, 887, 911, 929, 937, 953, 967, 977, 983

Offset

1,1

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Formula

x*(x+1)*(x+2)*(x+3) == 1 mod p, p is prime, 1 <= x <= p-4.

Example

p=7: 2*3*4*5=120 == 1 mod 7;
p=17: 2*3*4*5=120 == 1 mod 17 AND 12*13*14*15=32760 == 1 mod 17; for p=13: no triple == 1 mod 13;
p=23: 5*6*7*8 == 1 mod 23 AND 15*16*17*18== 1 mod 23 AND 19*20*21*22 == 1 mod 23; and so on. For the number of quadruples for a prime, see A256580.

Mathematica

fsiQ[n_]:=AnyTrue[Times@@@Partition[Range[n-1], 4, 1], Mod[#, n]==1&]; Select[ Prime[Range[200]], fsiQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 02 2019 *)

Prog

(R)
library(numbers)
IP <- vector()
t <- vector()
S <- vector()
IP <- c(Primes(1000))
for (j in 1:(length(IP))){
   for (i in 2:(IP[j]-4)){
       t[i-1] <- as.vector(mod((i*(i+1)*(i+2)*(i+3)), IP[j]))
       Z[j] <- sum(which(t==1))
       S[j] <- length(which(t==1))
   }
}
IP[S!=0]
#Carefully increase Primes(1000). It takes several hours for 100000.
(PARI) lista(nn) = forprime(p=2, nn, if (sum(x=1, p-4, ((x*(x+1)*(x+2)*(x+3)) % p) == 1) > 0, print1(p, ", "))); \\ Michel Marcus, Apr 03 2015

Crossrefs

Cf. A256567, A256580.

Sequence in context: A270951 A001132 A308816 * A165353 A048976 A088546

Adjacent sequences: A254675 A254676 A254677 * A254679 A254680 A254681

Keyword

nonn

Author

Marian Kraus, Apr 02 2015

Status

approved