Are there consecutive integers of the form $a^2b^3$ where $a$, $b$ > 1? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T20:40:28Z http://mathoverflow.net/feeds/question/33037 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33037/are-there-consecutive-integers-of-the-form-a2b3-where-a-b-1 Are there consecutive integers of the form $a^2b^3$ where $a$, $b$ > 1? Ken Fan 2010-07-23T03:51:30Z 2010-07-23T17:20:50Z <p>Let $S$ = { $a^2b^3$ : $a, b \in \mathbb{Z}_{>1}$ }.</p> <p>Does there exist $n$ such that $n$, $n+1 \in S$?</p> <p>Motivation: I was thinking about <a href="http://mathoverflow.net/questions/32412/question-on-consecutive-integers-with-similar-prime-factorizations" rel="nofollow">http://mathoverflow.net/questions/32412/question-on-consecutive-integers-with-similar-prime-factorizations</a>, wondering whether any such pair had to have prime signatures with at least one 1. This would follow if the answer to the above question is negative. (This would also follow from weaker versions of the above question too, such as taking out perfect $n$th powers from $S$.)</p> <p>Please note that $a$ and $b$ in the set definition are not allowed to be equal to 1. Otherwise, there'd be solutions like 8, 9 or 465124, 465125. (465124 = $(2\cdot 11 \cdot 31)^2$ and 465125 = $61^25^3$.)</p> http://mathoverflow.net/questions/33037/are-there-consecutive-integers-of-the-form-a2b3-where-a-b-1/33044#33044 Answer by Lavender Honey for Are there consecutive integers of the form $a^2b^3$ where $a$, $b$ > 1? Lavender Honey 2010-07-23T05:09:27Z 2010-07-23T05:09:27Z <p>As once remarked by Mahler, $x^2 - 8 y^2 = 1$ has infinitely many solutions with $27 | x$.</p> http://mathoverflow.net/questions/33037/are-there-consecutive-integers-of-the-form-a2b3-where-a-b-1/33114#33114 Answer by Max Alekseyev for Are there consecutive integers of the form $a^2b^3$ where $a$, $b$ > 1? Max Alekseyev 2010-07-23T17:20:50Z 2010-07-23T17:20:50Z <p>See also <a href="http://oeis.org/classic/A076445" rel="nofollow">http://oeis.org/classic/A076445</a> and this thread on the search for consecutive odd powerful numbers: <a href="http://www.mersenneforum.org/showthread.php?t=3474" rel="nofollow">http://www.mersenneforum.org/showthread.php?t=3474</a> Similar technique can be used for search for just consecutive powerful numbers (i.e., without the oddness restriction).</p> <p>P.S. And of course, <a href="http://oeis.org/classic/A060355" rel="nofollow">http://oeis.org/classic/A060355</a> is relevant.</p>