A co-ordinated search for primes in the Payam number series


These web pages co-ordinate the search for primes of the form k*2^n +/- 1, for special values of k, which I am dubbing Payam numbers.

Let the Payam number E{x} be the least value of k for which all prime numbers with order base 2 of less than or equal to x, are not factors of the series k*2^n+ or - 1 (from n=1 to infinity)

Reserve a range

See results

Certain values of k produce very few primes (sometimes zero primes!) for the series k*2^n+/-1 with n=1 to infinity.

An analysis of the factors of the numbers k*2^n +/-1 for increasing values of n show that certain factors occur frequently and regularly. For example, look at the table below, which looks at the factors of the series k*2^n+1 where k=1 and n increases.

Factors of k*2^n+1 (with k=1)
n
k*2^n+1 (with k=1)
Factors
1
3
3 (prime)
2
5
5 (prime)
3
9
3*3
4
17
17 (prime)
5
33
3*11
6
65
5*13
7
129
3*43
8
257
257 (prime)
9
513
3*3*3*19
10
1025
5*5*41
11
2049
3*683
12
4097
17*241

Several observations about the results extrapolated to n= infinity:

3 is a factor of every second n, or to put it in mathematical parlance, 3 has order 2, base 2 (the base is the 2 found in the formula k*2^n+1) and this is represented as e(3) = 2

5 appears as a factor of every fourth n, so 5 has order 4, base 2 ,or e(5) = 4

17 appears as a factor of every eighth n, so e(17)=8

We also find that e(11) = 10.....e(13) = 12.....e(17) = 8.....e(19) = 18.....e(41) = 20.....e(43) = 14 .....e(241) = 24.....e(257) = 16.....e(683)= 22 etc.

What is also interesting is that 7 and 23 (and maybe some other primes) never appear in this series as factors, when k=1. However if k=3 for example, then we would find that e(7) =3, and we would also note that 3 and 23 never appear as factors.

When k=5, we find that 3 and 17 are never factors, but 23 is, and e(23) = 11

Several assertions might be made here.

First, there seems little rhyme or reason why certain primes never appear as factors for certain values of k,

but when they do, they appear with the same e values,

but they do not appear predictable.

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Address questions about this web page to: Robert Smith

Last updated 9 November 2002