The Grand Internet Obstinate Number Search



In 1848, Camille Armand Jules Marie, better known as the "Prince de Polignac," conjectured that every odd number is the sum of a power of two and a prime. (For example 13 = 2**3+5.) He claimed to have proved this to be true for all numbers up to three million, but de Polignac probably would have kicked himself if he had know that he missed 127, which leaves residuals of 125, 123, 119, 111, 95 and 63 (all composites) when the possible powers of two are subtracted from it. There are another 16 of these odd numbers -- which my colleague Andy Edwards calls "obstinate numbers" -- that are less than 1000. There are an infinity of obstinate numbers greater than 1000. Most obstinate numbers we have discovered are prime themselves. The first composite obstinate number is 905.

What is the largest obstinate number you can compute?

Can you find an obstinate number terminating in every digit from 0 to 9, or are certain terminal digits impossible to find?

Other unanswered questions:

1) What is the smallest difference between adjacent obstinates?
2) Do any obstinates undulate? (Undulating numbers are of the form: ababababab.... For example, 171717 and 28282 are undulating numbers, but they're not obstinate, as far as I know.)
3) How are obstinates distributed through the numbers as we scan ever larger numbers.

I'll try to list world-record holders at my web site and a future book. If you make discoveries, give us some ideas about the kinds of search programs you used. 

From: "Daniel Dockery"  
> What is the largest obstinate number you can
> compute?  (Obstinates are defined below.)

The largest? I suppose it depends on your processing
power, and your patience in searching for them. : )
I stopped my search at:


  99999999999999999999999999999999999999999999 \
  99999999999999999999999999999999999999999999 \
  99999999999999999999999999999999999999999999 \
  99999999999999999999999999999999999999999999 \
  99999999037
It leaves a composite residue for all 621 possible powers of 2 that can be subtracted from it. If desired, I can send the list of them to whomever's interested (I chose not to post them here since it's a large list of large numbers). 
> Can you find an obstinate number terminating
> in every digit from 0 to 9, or are certain
> terminal digits impossible to find?

Obstinate numbers, since they are based on C.A.J.
Marie's conjecture, must be odd numbers, correct?
Terminating even digits, if so, would be impossible.
All the odd digits seem represented, though (e.g.,
1: 251, 331, 701; 3: 373, 1243, 1783;
5: 905, 1985, 2465; 7: 127, 337, 757;
9: 149, 509, 599, etc.)

> 1) What is the smallest difference between
> adjacent obstinates?

905 and 907 are both obstinate by this definition,
and have a difference of 2; since the numbers must
be odd, that's the smallest possible.

> 2) Do any obstinates undulate? (Undulating numbers
> are of the form: ababababab....  For example,
> 171717 and 28282 are undulating numbers, but
> they're not obstinate, as far as I know.)

6161
14141
39393
91919
1313131
1818181
7070707
7474747
7676767
7979797
59595959
73737373
343434343
757575757
797979797
929292929
1717171717
3131313131
9191919191
12121212121
14141414141
18181818181
32323232323
54545454545
78787878787
91919191919
7171717171717
25252525252525
29292929292929
37373737373737
43434343434343
67676767676767
97979797979797

I suppose there are others, but I only let the search
run for a bit.

> If you make discoveries, give us some ideas about
> the kinds of search programs you used.
Not sure if any of the above counts as a discovery, but as far as search programs, I found all of the above with simple scripts written in Maple V. --



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