A generated Sierpinski/Riesel problem

[sweety439]

The number of iterations (x -> a*x+b) until a prime is found, starting with n; or 0 if a prime is never found. (where a, b, n are integers, a > 1 (the a = 1 case can be proven by Dirichlet's theorem), b != 0, n > 0, gcd(a,b) = 1, gcd(n,b) = 1)

The b=a-1 case is the standard Riesel problem, and the b=-(a-1) case is the standard Sierpinski problem.

This is a list for all CK (least k coprime to b such that no prime in the iterations (x -> a*x+b)) for all (a,b) pairs such that 2<=a<=32, |b|<=32, gcd(a,b)=1

Format:

a, b: CK

(Searched up to 10^6, lists "0" if the CK is >10^6)

Of course there are some k making a full covering set with all or partial algebraic factors (e.g. k=5 and k=8 in the "4x+1" iteration), they are excluded from the conjectures.

Some CK's not in the text file were found:

the CK for 5x-2 is 1213059
the CK for 8x+3 is 1079770
the CK for 8x-3 is 5389955
the CK for 11x+6 is 1669093

[sweety439]

These are the text files for 2x+-3 for x<=1024 (gcd(x,3) =1), note that for 2x-3, this x must be >3, or the numbers would be negative.

For 3x+1 see https://www.rose-hulman.edu/~rickert/Compositeseq/, CK=6059 and there are 58 k's remain.

For 3x-1 (CK=5524), the status given in the text file.

4x+-1 are both proven.

4x+1: CK = 120, covering set: {3, 5, 7, 13} (k = 1, 5, 8, 16, 21, 33, 40, 56, 65, 85, 96 proven composite by full algebraic factors)

4x-1: CK = 140, covering set: {3, 5, 7, 13}

[sweety439]

Originally Posted by sweety439

The number of iterations (x -> a*x+b) until a prime is found, starting with n; or 0 if a prime is never found. (where a, b, n are integers, a > 1 (the a = 1 case can be proven by Dirichlet's theorem), b != 0, n > 0, gcd(a,b) = 1, gcd(n,b) = 1)

The b=a-1 case is the standard Riesel problem, and the b=-(a-1) case is the standard Sierpinski problem.

This is a list for all CK (least k coprime to b such that no prime in the iterations (x -> a*x+b)) for all (a,b) pairs such that 2<=a<=32, |b|<=32, gcd(a,b)=1

Format:

a, b: CK

(Searched up to 10^6, lists "0" if the CK is >10^6)

Of course there are some k making a full covering set with all or partial algebraic factors (e.g. k=5 and k=8 in the "4x+1" iteration), they are excluded from the conjectures.

Some CK's not in the text file were found:

the CK for 5x-2 is 1213059
the CK for 8x+3 is 1079770
the CK for 8x-3 is 5389955
the CK for 11x+6 is 1669093

e.g. for 3x+1:

1 --> 4 --> 13 (prime)
2 --> 7 (prime)
3 --> 10 --> 31 (prime)
4 --> 13 (prime)
5 --> 16 --> 49 --> 148 --> 445 --> 1336 --> 4009 --> 12028 --> 36085 --> 108256 --> 324769 --> 974308 --> 2922925 --> 8768776 --> 26306329 --> 78918988 --> 236756965 --> 710270896 --> 2130812689 --> 6392438068 --> 19177314205 --> 57531942616 --> 172595827849 (prime)
6 --> 19 (prime)
7 --> 22 --> 67 (prime)
8 --> 25 --> 76 --> 229 (prime)
9 --> 28 --> 85 --> 256 --> 769 (prime)
10 --> 31 (prime)
etc.

For 3x-1:

1 --> 2 (prime)
2 --> 5 (prime)
3 --> 8 --> 23 (prime)
4 --> 11 (prime)
5 --> 14 --> 41 (prime)
6 --> 17 (prime)
7 --> 20 --> 59 (prime)
8 --> 23 (prime)
9 --> 26 --> 77 --> 230 --> 689 --> 2066 --> 6197 (prime)
10 --> 29 (prime)
etc.