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 2022-06-23, 20:03 #1 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 5×233 Posts 10^a+1 is prime at least twice Hi again everybody, I'm curiouse about positibe integers of the form 10^a + 1 where a is a positive integer. see my code. > print("prime factorization of numbers of the form 10^a + 1"); for a to 20 do b := 10^a+1; c := ifactor(10^a+1); print(b, " factors as ", c) end do; "prime factorization of numbers of the form 10^a + 1" 11, " factors as ", (11) 101, " factors as ", (101) 1001, " factors as ", (7) (11) (13) 10001, " factors as ", (73) (137) 100001, " factors as ", (11) (9091) 1000001, " factors as ", (101) (9901) 10000001, " factors as ", (11) (909091) 100000001, " factors as ", (17) (5882353) 1000000001, " factors as ", (7) (11) (13) (19) (52579) Now that you know that One Billion has prime factorization of 7*11*13*19*52579, your life is complete. Another calculation shows that. 10^a + 1 is a composite number for integers 33. But a pattern does not make a theorem. Anyone care to take the calculation furthur? Regards, Matt PS Has someone else already done this calculation? Can we look this up somewhere? Cheers, Matt P.S. After nearly a 5 hour calculation, and I trust my computer for accurate results, I can say with certainty that the positive integers of the form 10^a+1 are composite for integers 'a' with 3 <= a <= 97. This I am sure of. I'm sure someone else could extend the dataset. Regards, Matthew Charles Anderson PPS Here is my raw Maple code > restart*print("prime factorization of numbers of the form 10^a + 1"); for a to 1000 do b := 10^a+1; c := ifactor(10^a+1); if isprime(b) then print(b, " factors as ", c) end if end do; "prime factorization of numbers of the form 10^a + 1" restart () 11, " factors as ", (11) 101, " factors as ", (101) Warning, computation interrupted (a checkpoint) a 97 #interesting_mathematical_trivia So to reiterate, 11 is prime, 101 is prime, and 10^a + 1 is not prime for a= 3,4,5,...,97. Have a nice day.
 2022-06-23, 20:34 #2 paulunderwood     Sep 2002 Database er0rr 22·1,063 Posts 10^2^k+1 is a Generalized Fermat Number. There are very few Fermat Numbers that are prime. There will be fewer base 10 GFN primes is my guess. Last fiddled with by paulunderwood on 2022-06-23 at 20:47
 2022-06-23, 20:49 #3 ATH Einyen     Dec 2003 Denmark 1101000101012 Posts All odd a>=3 is divisible by 11: a=2n+1 102n+1 + 1 = 100n * 10 + 1 = 1n * 10 + 1 = 0 (mod 11) All a=4n+2 >=6 is divisible by 101: 104n+2 + 1 = 10000n * 100 + 1 = 1n * 100 + 1 = 0 (mod 101) All a=8n+4 >=4 is divisible by 73: 108n+4 + 1 =100000000n * 10000 + 1 = 1n * 10000 + 1 = 0 (mod 73) There are probably several more of these. Last fiddled with by ATH on 2022-06-23 at 20:51
 2022-06-23, 21:03 #4 kar_bon     Mar 2006 Germany 3·23·43 Posts 5 hours of energy which could be saved by searching a well known database of such formats of number, see Studio Kamada for factors of 10^n+1 for n=1 to 150,000. FactorDB also contains those factorizations already.
 2022-06-23, 21:52 #5 charybdis     Apr 2020 797 Posts If k is odd, then x^k+1 factorizes as (x+1)(x^(k-1)-x^(k-2)+...-x+1). Substituting 10^m for x, we see that 10^km+1 is divisible by 10^m+1 if k is odd. In other words, 10^n+1 cannot be prime if n has an odd factor greater than 1. The only values of n for which this is not the case are powers of 2, so these are the only exponents for which it is possible for 10^n+1 to be prime. They are Generalized Fermat numbers as Paul says. See Wilfrid Keller's page for the status of the base 10 GFNs. No primes are known apart from 11, and it is almost certain that no more exist. 10^(2^31)+1 is the smallest with unknown status.

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