A218483 Fermat pseudoprimes to base 2 which are congruent to 1 (mod 8).

561, 1105, 1729, 1905, 2465, 4033, 4369, 4681, 6601, 8321, 8481, 10585, 11305, 12801, 15841, 16705, 18705, 18721, 23001, 23377, 25761, 30121, 30889, 31417, 31609, 33153, 34945, 39865, 41041, 41665, 46657, 52633, 62745, 65281, 74665, 75361, 83665, 85489

Offset: 1

Author

Marius Coman, Oct 30 2012

Keywords

• nonn

Comments

Old name was: Fermat pseudoprimes to base 2 of the form 8*p*n + p^2, where p is prime and n natural.

For p = 5 the formula becomes 40*n + 25. From the first 15 pseudoprimes divisible by 5, 12 are of the form 40*n + 25 (beside 3 of them which are of the form 40*n + 5). Conjecture: there are no pseudoprimes to base 2 of the form 40*n + 15.

Note: it can be seen that a pseudoprime can be written in this formula in more than one way: e.g., 561 = 8*3*23 + 3^2 = 8*11*5 + 11^2 = 8*17*2 + 17^2 or 1905 = 8*3*79 + 3^2 = 8*5*47 + 5^2.

Conjecture: If a Fermat pseudoprime to base 2 can be written as 8*p*n + p^2, where n is an integer and p one of its prime factors, then it can be written this way for any of its prime factors. Checked for all pseudoprimes from the sequence above.

Conjecture: If a Fermat pseudoprime to base 2 with two prime factors can be written as 8*p1*n + p1^2, where n is a natural number and p1 one of its two prime factors, then it can also be written as 8*p2*(-n) + p2^2, where p2 is the other prime factor. Checked for 4033 = 37*109(n = 9), 4369 = 17*257(n = 30), 4681 = 31*151(n = 15), 8321 = 53*157(n = 13), 18721 = 97*193(n = 12), 23377 = 97*241(n = 18), 31417 = 89*353(n = 33), 31609 = 73*433 (n = 45), 65281 = 97*673(n = 72), 85489 = 53*1613 (n = 195).

Conjecture: If a Fermat pseudoprime to base 2 cannot be written as 8*p*n + p^2, where n is an integer and p one of its prime factors, then it cannot be written this way for any of its prime factors. Checked for the following pseudoprimes: 341, 645, 1387, 2047, 2701, 2821, 3277, 4371, 5461, 7957, 10261, 13741, 13747, 13981, 14491, 15709, 19951, 29341, 31621, 42799, 49141, 49981, 55245, 60701, 60787, 63973, 65077, 68101, 72885, 80581, 83333.

Note: from the first 72 pseudoprimes, 39 can be written this way.

All three conjectures are true (obvious from new characterization). - Charles R Greathouse IV, Dec 07 2014

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Fermat Pseudoprime.
• Eric Weisstein's World of Mathematics, Poulet Number.

Crossrefs

Cf. A001567, A216023, A214305.

Programs

Maple

select(t -> 2 &^ t mod t = 2 and not isprime(t), [seq(1+8*j,j=0..10^5)]); # Robert Israel, Dec 07 2014

Mathematica

Select[8 * Range[10^4] + 1, PowerMod[2, # - 1, #] == 1 && CompositeQ[#] &] (* Amiram Eldar, Mar 30 2021 *)

PARI

is(n)=n%8==1 && Mod(2,n)^n==2 && !isprime(n) \\ Charles R Greathouse IV, Dec 07 2014

Extensions

Corrected by Charles R Greathouse IV, Dec 07 2014

New name from Charles R Greathouse IV, Dec 07 2014