## 25478Re: [PrimeNumbers] sum of two p-smooth numbers

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• Feb 15, 2014
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Good find!  And, your copy and paste looks accurate.

From the AMS paper I see that starting at prime = 31 is a first occurrence where the least number solution is not prime.   In that case, the least number solution is  307271 = 109 • 2819, and the least prime solution is 366791.
Out of curiosity I decided to continue with my investigation past prime 19, up to prime 31, because I was curious.  Namely, would the least number that can't be written as the sum of two 31-smooth numbers correspond to the least number solution (307271) or the least prime solution (366791) ?
It turns out, neither!
The first 5 integers > 1 that can't be written as the sum of two 31-smooth numbers are

118271, 236542, 307271, 354813, 366791.

118281 = 101*1171.

Go figure...

Mark

From
D. H. Lehmer, Emma Lehmer and Daniel Shanks
http://www.ams.org/mcom/1970-24-110/S0025-5718-1970-0271006-X
R. F. Lukes, C. D. Patterson and H. C. Williams
http://www.ams.org/mcom/1996-65-213/S0025-5718-96-00678-3
it seems that the continuation of
http://oeis.org/A002223/b002223.txt
is as follows:
27 33857579279
28 89206899239
29 121560956039
30 328878692999
31 328878692999
32 513928659191
33 4306732833311
34 4306732833311
35 4306732833311
36 8402847753431
37 70864718555231
38 317398900373231
39 501108392233679
40 501108392233679
41 501108392233679
42 501108392233679
43 5551185799073591
44 5551185799073591
45 5551185799073591
46 7832488789769159
47 7832488789769159
48 102097158739597271
49 102097158739597271
50 315759454565514431
51 868116409360316399
52 3412527725201978759
53 3546374752298322551
54 3546374752298322551
55 3546374752298322551
56 3546374752298322551
57 3546374752298322551
However this needs checking by someone
better at cutting and pasting than I.
David (struggling with abominable yahoo interface)
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