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"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).> 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

> 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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