

A215326


Fermat pseudoprimes to base 2 of the form m*n^2 + (11*m  23)*n + 19*m  49, where m, n >= 0.


1



341, 645, 1105, 1387, 2047, 2465, 2821, 3277, 4033, 5461, 6601, 7957, 8321, 11305, 13747, 15841, 16705, 19951, 23001, 25761, 30889, 31417, 31621, 39865, 41665, 49981, 65077, 68101, 74665, 83333, 83665, 85489, 88357, 90751, 107185, 137149, 158369
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OFFSET

1,1


COMMENTS

Fermat pseudoprimes to base 2 are also called Poulet numbers.
The solutions (m,n) for the Poulet numbers from the sequence above are (3,9); (3,13); (4,14); (4,16); (9,11) and (4,20); (3,27); (3,29); (4,26); (3,35); (290,0) and (3,41); (350,0) and (4,38); (259,1); (3,51); (367,1); (4,56); (94,8) and (3,71); (4,62); (329,3) and (4,68); (379,3); (3,91); (182,8); (319,5) and (4,86); (3,101); (888,2); (928,2) and (66,20); (43,29); (659,5); (3,149); (438,8) and (4,134); (3,165); (4406,0) and (4,142); (4502,0); (4,146); (4,148); (2384,2) and (38,48); (1387,5); (5111,1).
Note: for n = 2 the formula becomes (m  3), and for n = 9 it becomes (m + 158), so all the Poulet numbers have at least these integer solutions to this formula.
Another interesting observation about the integer solutions: for n = 1 the formula becomes (9*m  26), and 37 of the first 100 Poulet numbers can be written this way! This means that, for more than a third of Poulet numbers P checked, it is true that (P + 8) is divisible by 9 (for comparison, this relation is true for just 14 of the first 100 primes).


LINKS



EXAMPLE

For m = 3 the formula is 3*n^2 + 10*n + 8, giving the following Poulet numbers: 341, 645, 2465, 2821, 4033, 5461, 8321, 15841, 25761, 31621, 68101, 83333, etc.
For m = 4 the formula is 4*n^2 + 21*n + 27, giving the following Poulet numbers: 1105, 1387, 2047, 3277, 6601, 13747, 16705, 19951, 31417, 83665, 88357, 90751, etc.


PROG

(PARI) is(n)=if(Mod(2, n)^(n1)!=1isprime(n), return(0)); for(k=0, sqrtint(n+66)+6, if(Mod(n, k^2+11*k+19)+23*k+49==0, return(1))); 0 \\ Charles R Greathouse IV, Dec 07 2014


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



