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A068435
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Consecutive prime powers without a prime between them.
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7
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OFFSET
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1,1
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COMMENTS
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Only 5 pairs are known up to 4*10^18. Legendre's conjecture states that there is a prime number between n^2 and (n + 1)^2 for every positive integer n. The conjecture has been verified up to n = 2*10^9. So to that bound we only have to check for two prime powers where at least one has an exponent of at least 3. That has been done to prime powers <= 10^22.
If there is another pair besides the first five listed with both numbers <= 10^22 then Legendre's conjecture is false.
Proof: If there is another such pair with both numbers <= 10^22 then it must be of the form [p^2, q^2] where p is a prime and q is the least prime larger than p. Then q - p >= 2 (as p != 2). So there is no prime between p^2 and q^2 and hence there is no prime between p^2 and (p+1)^2. This is a counterexample to Legendre's conjecture. (End)
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LINKS
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EXAMPLE
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8 = 2^3, 9 = 3^2, there is no prime between 8 and 9.
25 = 5^2, 27 = 3^3, there is no prime between 25 and 27.
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MATHEMATICA
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With[{upto=33000}, Select[Partition[Select[Range[upto], PrimePowerQ], 2, 1], NoneTrue[#, PrimeQ]&]] (* Paolo Xausa, Oct 29 2023 *)
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PROG
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(PARI) ispp(x) = !isprime(x) && isprimepower(x);
lista(nn=50000) = {my(prec = 0); for (i=1, nn, if (ispp(i), if (! prec, prec = i, if (primepi(i) == primepi(prec), print1(prec, ", ", i, ", ")); prec = i; ); ); ); } \\ Michel Marcus, Aug 24 2019
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CROSSREFS
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Cf. A116086 and A116455 (for perfect powers, but not necessarily prime powers).
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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