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%I #29 Jul 25 2019 08:36:13
%S 5,41,281,1849,9007,35803
%N a(n) is the least integer not the difference of two prime(n)-smooth numbers.
%C The known terms have been found by exhaustive search and then proved not to be the difference of prime(n)-smooth numbers using assertions such as <a,b,...> +- a(n) !== <c,d,...> (modulo m) meaning that no element of the subgroup of Z/m generated by a,b,... added to a(n) is congruent modulo m with an element of the subgroup generated by <c,d,...>. For example: <2> +- 41 !== <3> (mod 91) and the fact that 41+1 is not 3-smooth is enough to prove that 41 is not the difference of 3-smooth numbers; <2> + 281 !== <3,5> (mod 13981), <2> - 281 !== <3,5> (mod 76627) and <3> +- 281 !== <2,5> along with the fact that 281+1 is not 5-smooth is enough to show that 281 is not the difference of 5-smooth numbers. The proofs get exponentially harder as n increases. For example, <2, 11> + 9007 !== <3, 5, 7> (mod 308859288230831), or <2,5,7> + 35803 !== <3,11,13> (mod 2219897250633559197203).
%C The next few terms are conjectured to be 158857, 681179, 2516509, 10772123, 51292187, 186323681; if they were not, they would provide examples of ABC-triples with quality greater than 2.
%H Esteban Crespi de Valldaura, <a href="/A308247/a308247.txt">Proof that a(n) is not prime(n)-smooth for n=2,3,4,5,6</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Abc_conjecture#Highest-quality_triples">abc conjecture</a>
%e We see that 1 = 2-1, 2 = 4-2, 3 = 4-1, and 4 = 8-4. It is easy to see that 5 is not the difference of two powers of 2, so a(1) = 5. In the same way we can see that all the integers up to 40 are the difference of 3-smooth numbers, but as shown above 41 is not, so a(2)=41.
%Y P-smooth_numbers: A000079, A003586, A051037, A002473, A051038, ...
%Y a(i) is the first term in each of A101082, A290365, A308456, A326318, A326319, A326320.
%K nonn,more
%O 1,1
%A _Esteban Crespi de Valldaura_, May 16 2019
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