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A068435 Consecutive prime powers without a prime between them. 4
8, 9, 25, 27, 121, 125, 2187, 2197, 32761, 32768 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

From David A. Corneth, Aug 24 2019: (Start)

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)

LINKS

Table of n, a(n) for n=1..10.

David A. Corneth, PARI program to search for pairs where not both numbers are squares of primes

Wikipedia, Legendre's conjecture

EXAMPLE

8=2^3, 9=2^3, there is no prime between 8 and 9.

25=5^2, 27=3^3, there is no prime between 25 and 27.

PROG

(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

CROSSREFS

Cf. A014085, A025475, A060846, A067871, A246547.

Cf. A116086 and A116455 (for perfect powers, but not necessarily prime powers).

Sequence in context: A109097 A335770 A115645 * A302554 A124438 A124184

Adjacent sequences:  A068432 A068433 A068434 * A068436 A068437 A068438

KEYWORD

nonn,more

AUTHOR

Jon Perry, Mar 09 2002

STATUS

approved

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Last modified July 6 04:46 EDT 2020. Contains 335475 sequences. (Running on oeis4.)