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A positive integer $n$ is called powerful (OEIS: A001694) if $p^2$ divides $n$ whenever $p$ is a prime that divides $n$. Equivalently, $n$ is powerful if $n = a^2b^3$, where $a$ and $b$ are positive integers.

Conjecture: If $n$ is a powerful integer greater than one, then there is a prime $p$ and a powerful number $m$ such that $n=m+p$.

This conjecture holds for every powerful number $\leq 1000$. Given its similarity to the Goldbach conjecture, my suspicion is that this might be quite difficult to prove. Any references/ideas/partial results on this would be greatly appreciated.

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    $\begingroup$ @Wojowu By convention, $1$ is considered powerful, and since $4-1=3$ and $8-1=7$, the conjecture holds for these values. $\endgroup$ – Pietro Paparella May 5 '17 at 17:02
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    $\begingroup$ Confirmed up to $10^6$ (using SageMath). $\endgroup$ – Wojowu May 5 '17 at 17:18
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    $\begingroup$ Sequence A133364 is relevant. Every positive integer $< 10^7$ except $1$, $2$ and $5$ can be written as $m + p$ with $m$ powerful and $p$ prime. The obvious conjecture is that this is true for all positive integers except $1$, $2$ and $5$. The positive integers $n < 10^7$ for which $m+p = n$ has exactly one solution are $3, 4, 7, 8, 9, 10, 13, 16, 17, 22, 24, 25, 26, 31, 36, 58, 64, 76, 82, 120, 170, 193, 196, 214, 324, 328, 370, 412, 562, 676, 730, 10404$. $\endgroup$ – Robert Israel May 5 '17 at 18:13
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    $\begingroup$ The fact that so large a number as $10404$ manages to be the sum of a powerful number and a prime in only one way ($343 + 10061$) makes me suspect that this conjecture, if true, may be even harder than Goldbach. $\endgroup$ – Robert Israel May 5 '17 at 21:56
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    $\begingroup$ Hardy and Littlewood conjectured (in "Partitio Numerum III") that every sufficiently large non-square is the sum of a square and a prime. This would settle all cases except for small powerful numbers and large square numbers. (Not that this conjecture is anywhere close to being resolved; as noted above, it is probably harder than the even Goldbach conjecture.) $\endgroup$ – Terry Tao May 7 '17 at 21:43

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