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%I #53 Nov 13 2023 05:20:05
%S 8,9,25,27,121,125,2187,2197,32761,32768
%N Consecutive prime powers without a prime between them.
%C From _David A. Corneth_, Aug 24 2019: (Start)
%C 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.
%C If there is another pair besides the first five listed with both numbers <= 10^22 then Legendre's conjecture is false.
%C 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)
%H David A. Corneth, <a href="/A068435/a068435.gp.txt">PARI program to search for pairs where not both numbers are squares of primes</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Legendre%27s_conjecture">Legendre's conjecture</a>
%e 8 = 2^3, 9 = 3^2, there is no prime between 8 and 9.
%e 25 = 5^2, 27 = 3^3, there is no prime between 25 and 27.
%t With[{upto=33000},Select[Partition[Select[Range[upto],PrimePowerQ],2,1],NoneTrue[#,PrimeQ]&]] (* _Paolo Xausa_, Oct 29 2023 *)
%o (PARI) ispp(x) = !isprime(x) && isprimepower(x);
%o 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
%Y Cf. A014085, A025475, A060846, A067871, A068315, A246547, A366835.
%Y Cf. A116086 and A116455 (for perfect powers, but not necessarily prime powers).
%K nonn,more
%O 1,1
%A _Jon Perry_, Mar 09 2002
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