Quote:
Originally Posted by sweety439
Like Bunyakovsky conjecture, it is conjectured that for all integer triples (k, b, c) satisfying these conditions:
1. k>=1, b>=2, c != 0
2. gcd(k, c) = 1, gcd(b, c) = 1
3. there is no finite set {p_1, p_2, p_3, ..., p_u} (all p_i (1<=i<=u) are primes) and finite set {r_1, r_2, r_3, ..., r_s} (all r_i (1<=i<=s) are integers > 1) such that for every integer n>=1:
either
(k*b^n+c)/gcd(k+c, b1) is divisible by at least one p_i (1<=i<=u)
or
k*b^n and c are both r_ith powers for at least one r_i (1<=i<=s)
or
one of k*b^n and c is a 4th power, another is of the form 4*t^4 with integer t
4. the triple (k, b, c) is not in this case: c = 1, b = q^m, k = q^r, where q is an integer not of the form t^s with odd s > 1, and m and r are integers having no common odd prime factor, and the exponent of highest power of 2 dividing r >= the exponent of highest power of 2 dividing m, and the equation 2^x = r (mod m) has no solution
Then there are infinitely many integers n>=1 such that (k*b^n+c)/gcd(k+c, b1) is prime.

Example of some (k,b,c) triple (k>=1, b>=2, c != 0, gcd(k, c) = 1, gcd(b, c) = 1) not satisfying these conditions:
* (k,b,c) = (78557,2,1), in which all numbers are divisible by at least one of 3, 5, 7, 13, 19, 37, 73
* (k,b,c) = (271129,2,1), in which all numbers are divisible by at least one of 3, 5, 7, 13, 17, 241
* (k,b,c) = (11047,3,1), in which all numbers are divisible by at least one of 2, 5, 7, 13, 73
* (k,b,c) = (419,4,1), in which all numbers are divisible by at least one of 3, 5, 7, 13
* (k,b,c) = (659,4,1), in which all numbers are divisible by at least one of 3, 5, 13, 17, 241
* (k,b,c) = (794,4,1), in which all numbers are divisible by at least one of 3, 5, 7, 13
* (k,b,c) = (7,5,1), in which all numbers are divisible by either 2 or 3
* (k,b,c) = (11,5,1), in which all numbers are divisible by either 2 or 3
* (k,b,c) = (174308,6,1), in which all numbers are divisible by at least one of 7, 13, 31, 37, 97
* (k,b,c) = (47,8,1), in which all numbers are divisible by at least one of 3, 5, 13
* (k,b,c) = (989,10,1), in which all numbers are divisible by at least one of 3, 7, 11, 13
* (k,b,c) = (5,11,1), in which all numbers are divisible by either 2 or 3
* (k,b,c) = (7,11,1), in which all numbers are divisible by either 2 or 3
* (k,b,c) = (521,12,1), in which all numbers are divisible by at least one of 5, 13, 29
* (k,b,c) = (4,14,1), in which all numbers are divisible by either 3 or 5
* (k,b,c) = (509203,2,1), in which all numbers are divisible by at least one of 3, 5, 7, 13, 17, 241
* (k,b,c) = (334,10,1), in which all numbers are divisible by at least one of 3, 7, 13, 37
* (k,b,c) = (1585,10,1), in which all numbers are divisible by at least one of 3, 7, 11, 13
* (k,b,c) = (376,12,1), in which all numbers are divisible by at least one of 5, 13, 29
* (k,b,c) = (919,4,1), in which all numbers are divisible by at least one of 3, 5, 7, 13
* (k,b,c) = (13,5,1), in which all numbers are divisible by either 2 or 3
* (k,b,c) = (17,5,1), in which all numbers are divisible by either 2 or 3
* (k,b,c) = (14,8,1), in which all numbers are divisible by at least one of 3, 5, 13
* (k,b,c) = (116,8,1), in which all numbers are divisible by at least one of 3, 5, 13
* (k,b,c) = (148,8,1), in which all numbers are divisible by at least one of 3, 5, 13
* (k,b,c) = (5,11,1), in which all numbers are divisible by either 2 or 3
* (k,b,c) = (7,11,1), in which all numbers are divisible by either 2 or 3
* (k,b,c) = (1,4,1), in which all numbers factored as difference of squares
* (k,b,c) = (9,4,1), in which all numbers factored as difference of squares
* (k,b,c) = (1,9,1), in which all numbers factored as difference of squares
* (k,b,c) = (4,9,1), in which all numbers factored as difference of squares
* (k,b,c) = (16,9,1), in which all numbers factored as difference of squares
* (k,b,c) = (1,16,1), in which all numbers factored as difference of squares
* (k,b,c) = (4,16,1), in which all numbers factored as difference of squares
* (k,b,c) = (9,16,1), in which all numbers factored as difference of squares
* (k,b,c) = (25,16,1), in which all numbers factored as difference of squares
* (k,b,c) = (36,16,1), in which all numbers factored as difference of squares
* (k,b,c) = (1,4,9), in which all numbers factored as difference of squares
* (k,b,c) = (1,4,25), in which all numbers factored as difference of squares
* (k,b,c) = (1,9,4), in which all numbers factored as difference of squares
* (k,b,c) = (1,9,16), in which all numbers factored as difference of squares
* (k,b,c) = (1,4,25), in which all numbers factored as difference of squares
* (k,b,c) = (1,8,1), in which all numbers factored as sum of cubes
* (k,b,c) = (27,8,1), in which all numbers factored as sum of cubes
* (k,b,c) = (125,8,1), in which all numbers factored as sum of cubes
* (k,b,c) = (343,8,1), in which all numbers factored as sum of cubes
* (k,b,c) = (729,8,1), in which all numbers factored as sum of cubes
* (k,b,c) = (1,8,27), in which all numbers factored as sum of cubes
* (k,b,c) = (1,27,1), in which all numbers factored as sum of cubes
* (k,b,c) = (8,27,1), in which all numbers factored as sum of cubes
* (k,b,c) = (1,27,8), in which all numbers factored as sum of cubes
* (k,b,c) = (1,8,1), in which all numbers factored as difference of cubes
* (k,b,c) = (27,8,1), in which all numbers factored as difference of cubes
* (k,b,c) = (1,8,27), in which all numbers factored as difference of cubes
* (k,b,c) = (125,8,1), in which all numbers factored as difference of cubes
* (k,b,c) = (1,27,1), in which all numbers factored as difference of cubes
* (k,b,c) = (8,27,1), in which all numbers factored as difference of cubes
* (k,b,c) = (1,27,8), in which all numbers factored as difference of cubes
* (k,b,c) = (1,32,1), in which all numbers factored as sum of 5th powers
* (k,b,c) = (1,32,1), in which all numbers factored as difference of 5th powers
* (k,b,c) = (4,16,1), in which all numbers factored as x^4+4*y^4
* (k,b,c) = (324,16,1), in which all numbers factored as x^4+4*y^4
* (k,b,c) = (2500,16,1), in which all numbers factored as x^4+4*y^4
* (k,b,c) = (4,81,1), in which all numbers factored as x^4+4*y^4
* (k,b,c) = (4,256,1), in which all numbers factored as x^4+4*y^4
* (k,b,c) = (4,625,1), in which all numbers factored as x^4+4*y^4
* (k,b,c) = (64,81,1), in which all numbers factored as x^4+4*y^4
* (k,b,c) = (64,256,1), in which all numbers factored as x^4+4*y^4
* (k,b,c) = (64,625,1), in which all numbers factored as x^4+4*y^4
* (k,b,c) = (324,256,1), in which all numbers factored as x^4+4*y^4
* (k,b,c) = (324,625,1), in which all numbers factored as x^4+4*y^4
* (k,b,c) = (25,12,1), in which even n factored as difference of squares and odd n is divisible by 13
* (k,b,c) = (64,12,1), in which even n factored as difference of squares and odd n is divisible by 13
* (k,b,c) = (4,19,1), in which even n factored as difference of squares and odd n is divisible by 5
* (k,b,c) = (9,14,1), in which even n factored as difference of squares and odd n is divisible by 5
* (k,b,c) = (4,24,1), in which even n factored as difference of squares and odd n is divisible by 5
* (k,b,c) = (9,24,1), in which even n factored as difference of squares and odd n is divisible by 13
* (k,b,c) = (4,39,1), in which even n factored as difference of squares and odd n is divisible by 5
* (k,b,c) = (9,34,1), in which even n factored as difference of squares and odd n is divisible by 5
* (k,b,c) = (81,17,1), in which even n factored as difference of squares and odd n is divisible by 2
* (k,b,c) = (144,28,1), in which even n factored as difference of squares and odd n is divisible by 29
* (k,b,c) = (289,28,1), in which even n factored as difference of squares and odd n is divisible by 29
* (k,b,c) = (16,33,1), in which even n factored as difference of squares and odd n is divisible by 17
* (k,b,c) = (225,33,1), in which even n factored as difference of squares and odd n is divisible by 2
* (k,b,c) = (289,33,1), in which even n factored as difference of squares and odd n is divisible by 2
* (k,b,c) = (6,24,1), in which odd n factored as difference of squares and even n is divisible by 5
* (k,b,c) = (27,12,1), in which odd n factored as difference of squares and even n is divisible by 13
* (k,b,c) = (6,54,1), in which odd n factored as difference of squares and even n is divisible by 5
* (k,b,c) = (76,19,1), in which odd n factored as difference of squares and even n is divisible by 5
* (k,b,c) = (126,14,1), in which odd n factored as difference of squares and even n is divisible by 5
* (k,b,c) = (300,12,1), in which odd n factored as difference of squares and even n is divisible by 13
* (k,b,c) = (16,12,49), in which even n factored as difference of squares and odd n is divisible by 13
* (k,b,c) = (441,12,1), in which even n factored as difference of squares and odd n is divisible by 13
* (k,b,c) = (1156,12,1), in which even n factored as difference of squares and odd n is divisible by 13
* (k,b,c) = (25,17,9), in which even n factored as difference of squares and odd n is divisible by 2
* (k,b,c) = (1369,30,1), in which even n factored as difference of squares and odd n is divisible by at least one of 7, 13, 19
* (k,b,c) = (400,88,1), in which even n factored as difference of squares and odd n is divisible by at least one of 3, 7, 13
* (k,b,c) = (324,95,1), in which even n factored as difference of squares and odd n is divisible by at least one of 7, 13, 229
* (k,b,c) = (3600,270,1), in which even n factored as difference of squares and odd n is divisible by at least one of 7, 13, 37
* (k,b,c) = (93025,498,1), in which even n factored as difference of squares and odd n is divisible by at least one of 13, 67, 241
* (k,b,c) = (61009,540,1), in which even n factored as difference of squares and odd n is divisible by either 17 or 1009
* (k,b,c) = (343,10,1), in which n divisible by 3 factored as difference of cubes and other n divisible by either 3 or 37
* (k,b,c) = (3511808,63,1), in which n divisible by 3 factored as sum of cubes and other n divisible by either 37 or 109
* (k,b,c) = (27000000,63,1), in which n divisible by 3 factored as sum of cubes and other n divisible by either 37 or 109
* (k,b,c) = (64,957,1), in which n divisible by 3 factored as sum of cubes and other n divisible by either 19 or 73
* (k,b,c) = (2500,13,1), in which n divisible by 4 factored as x^4+4*y^4 and other n divisible by either 7 or 17
* (k,b,c) = (2500,55,1), in which n divisible by 4 factored as x^4+4*y^4 and other n divisible by either 7 or 17
* (k,b,c) = (16,200,1), in which n == 2 mod 4 factored as x^4+4*y^4 and other n divisible by either 3 or 17
* (k,b,c) = (64,936,1), in which even n factored as difference of squares and n divisible by 3 factored as difference of cubes and other n divisible by either 37 or 109
* (k,b,c) = (8,128,1), in which the form equals 2^(7*n+3)+1 but 7*n+3 cannot be power of 2
* (k,b,c) = (32,128,1), in which the form equals 2^(7*n+5)+1 but 7*n+5 cannot be power of 2
* (k,b,c) = (64,128,1), in which the form equals 2^(7*n+6)+1 but 7*n+6 cannot be power of 2
* (k,b,c) = (8,131072,1), in which the form equals 2^(17*n+3)+1 but 17*n+3 cannot be power of 2
* (k,b,c) = (32,131072,1), in which the form equals 2^(17*n+5)+1 but 17*n+5 cannot be power of 2
* (k,b,c) = (128,131072,1), in which the form equals 2^(17*n+7)+1 but 17*n+7 cannot be power of 2
* (k,b,c) = (27,2187,1), in which the form equals (3^(7*n+3)+1)/2 but 7*n+3 cannot be power of 2
* (k,b,c) = (243,2187,1), in which the form equals (3^(7*n+5)+1)/2 but 7*n+5 cannot be power of 2