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

On the first assertion, actually there is a pattern underlying when the factors will appear. This can be seen from the following table which varies k but keeps n constant. Let n = 5, say. The table shows the factors of k*2^5+1

Factors of k*2^5+1 (with k=1)
k
k*2^5+1
Factors
1
33
3*11
2
65
5*13
3
97
97
4
129
3*43
5
161
7*23
6
193
193
7
225
3*3*5*5
8
257
257
9
289
17*17
10
321
3*107
11
353
353
12
385
5*7*11

We can see that 3 appears every third k, 5 every 5th k, 7 every 7th k, 11 every 11th k. Phil Carmody points out that -1 is in k's coset of the multiplicative subgroup generated by 2

On the second assertion e values are constant, but there is a reason for their values. For those mathematicians amongst you:

For each prime p, we define the exponent e of p base 2 as the least integer n such that 2^n=1(mod p). Thus we have 2^e=1(p) which means p divides 2^e-1.

This looks horrible for people like me who do not spend their world in modular arithmetic, but perhaps the following table might help to see the pattern:

Factors of 2^n-1
k
2^n-1
Factors
1
1
1
2
3
3
3
7
7
4
15
3*5
5
31
31
6
63
3*3*7
7
127
127
8
255
3*5*17
9
511
7*73
10
1027
3*11*31
11
2047
23*89
12
4095
3*3*5*7*13

The bold numbers represent the first time each prime appears, and these are equivalent to the e numbers we saw above. Neat!

So?

Well, we saw that when k equals 1, then 3 and 17 are never factors. There are certain k values for which all of the primes which have a low e value never appear as factors of our series.

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Address questions about this web page to: Robert Smith

Last updated 9 November 2002