Wikipedia:Reference desk/Archives/Mathematics/2024 April 9

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April 9

What is the smallest positive integer which is known to not divide any (even or odd) perfect number?

What is the smallest positive integer which is known to not divide any (even or odd) perfect number? Also, how many positive integers <= 400 are known to not divide any (even or odd) perfect number? 49.217.123.95 (talk) 08:05, 9 April 2024 (UTC)

All positive integers divide a perfect number. Writing $n=2^{a}b,$ in which $b$ is odd, it divides a perfect number of the form $2^{p-1}(2^{p}-1)$ for $p\leq a+b.$  --Lambiam 17:03, 9 April 2024 (UTC)

Are you sure about that, Lambiam? We know of only 51 perfect numbers, and we do not know whether there are infinitely many of them. Assuming there is a highest one, there must be infinitely many numbers greater than that, which, by definition, do not divide any lower numbers. Or, is your formula somehow proof that there are indeed infinitely many? -- Jack of Oz ^{[pleasantries]} 19:22, 9 April 2024 (UTC)

No, thank you for correcting me. All positive integers divide a number (in fact, infinitely many) of the form $2^{p-1}(2^{p}-1),$ but these need not be perfect numbers.  --Lambiam 00:13, 10 April 2024 (UTC)

Check. :) -- Jack of Oz ^{[pleasantries]} 02:27, 10 April 2024 (UTC)

As far as I can tell, from our article on perfect numbers (in particular, the section on odd perfect numbers; even perfect numbers are precisely divisible by all odd primes of the form $2^{p}-1$, and assuming that there are infinitely many such primes, all powers of 2 as well), it would seem that the answer is 105. GalacticShoe (talk) 02:22, 10 April 2024 (UTC)

Obviously this gives a trivial lower bound of 3 on the number of positive integers at most 400 known to not divide any perfect number. GalacticShoe (talk) 02:23, 10 April 2024 (UTC)

Even numbers cannot be factors of odd perfect numbers, thus there should be a larger lower bound. 220.132.230.56 (talk) 21:11, 12 April 2024 (UTC)

Good point. The sequence of even numbers that divide no perfect numbers is $10,12,18,20,22,24,26,30,34,36,38,40,42,44,46,48,50...$, basically even numbers not of the form $2^{k}(2^{p}-1)$ where $2^{p}-1$ is prime and $1\leq k<p$. GalacticShoe (talk) 23:47, 12 April 2024 (UTC)

So how many positive integers <= 400 are known to not divide any (even or odd) perfect number? 220.132.230.56 (talk) 13:35, 13 April 2024 (UTC)

Is it possible to use all n-cubes with n <= 5 assembled into a 2*3*31 rectangular cuboid?

Is it possible to use all n-cubes with n <= 5 (reflections counted as distinct), with totally 186 cubes, assembled into a 2*3*31 rectangular cuboid?

Also, for which positive integer N, it is possible to use all n-cubes with n <= N (reflections counted as distinct), assembled into a rectangular cuboid? 2402:7500:916:7991:1CB3:D7F2:BD28:3F3A (talk) 08:49, 9 April 2024 (UTC)

There are 11-polycubes with holes in the center, and it's impossible to fill this hole with anything but a single cube. There are several such shapes, and you only get one single cube, so N≥11 is impossible. You run into similar problems with holes when you try to tile with heptominos. You could stipulate that the pieces with holes not be included, but that would be a different problem. For N=5, it seems to me that best (or at least the most fun) approach would be to whip up a set with a 3-D printer and experiment. Sometimes a parity argument can give a simple impossibility proof, but that seems unlikely in this case. --RDBury (talk) 12:56, 9 April 2024 (UTC)