Here's a non-solution, but maybe the idea can be tweaked to work: (x^6+6)^5 = x^30 + 30 x^24 + 360 x^18 + 2160 x^12 + ... (x^10+10x^4)^3 = x^30 + 30 x^24 + 300 x^18 + 1000 x^12 The square roots are x^15 + 15 x^9 + 67.5 x^3 +- O(1/x^3) and X^15 + 15 x^9 + 37.5 x^3 +- O(1/x^3) The square roots are too far apart (delta = 30 x^3, needs to be ~1). Tweak away! There's also a heuristic argument that there should be an infinite set of cubes and fifth powers close together: There are x^(2/5) fifth powers below x^2, and x^(2/3) cubes. The distributions are fairly uniform, so assuming randomness is plausible. The product of the number of cubes times the number of fifth powers is x^(16/15), which is bigger than x^1, so we may expect around x^(1/15) cases where a cube and fifth power are between consecutive squares. This fails for cubes & seventh powers since 2/3 + 2/7 < 1, and similarly for other odd power combinations. And it offers heuristic support for the idea that there may be only a finite number of triples --- maybe zero. In contrast to the situation with pesky primes, the various powers have smooth distributions, so it may be possible to prove something. Rich ----- Quoting Allan Wechsler <acwacw@gmail.com>:
A number b^e, where b is a positive integer, and e an integer at least 2, we will call a "nontrivial power". This term is from Neil Sloane, and I think it's superior to the term "perfect power" which seems to have some currency.
The sequence of perfect powers is collected as A001597 at OEIS.
We naturally turn our attention to the b's and e's of the entries in this sequence, but there is some ambiguity here, exemplified by 16 = 4^2 = 2^4. Let us, for the moment, resolve the ambiguity by preferring, for this purpose, the representation with the smallest exponent. With this choice, the sequence of b's is A072813, while the e's are at A072814.
The exponent sequence A072814 is dominated by 2's, because cubes and other higher powers are much rarer than squares. Our attention is attracted by the rare cases when two higher powers are sandwiched between consecutive squares. The smallest example is the pair 3^3 = 27 and 2^5 = 32, sandwiched between the squares of 5 and 6.
The next example occurs between the squares of 11 and 12, 5^3 = 125 and 2^7 = 128.
The third example has 3^7 = 2187 and 13^3 = 2197, sandwiched between the squares of 46 and 47.
Are there an infinite number of such examples?
Are there any examples of *three* higher powers between consecutive squares? Hugo Pfoertner and I have expressed cautious skepticism; if I read Hugo's tone right, he and I both suspect there aren't any. Perhaps it is too ambitious to call this a conjecture, as I did in the subject of this message.
Similar questions about the existence of primes in intervals of various sizes are notoriously hard, but I would expect this problem to be much easier. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun