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teliot
teliot
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October 14th, 2021 at 12:24:56 PM permalink
Quote: ThatDonGuy

Quote: ThatDonGuy

Quote: teliot

For any integer N, consider the list of all primes that are in the half-open interval (N/2,N]. Call them p1, p2, p3 ... , pk. The question is, for which N is N!/(p1*p2*p3*...*pk) a perfect square? In other words, divide N! by all the primes greater than N/2 and less than or equal to N (the Bertrand's Postulate primes) and ask if what's leftover is a perfect square.

Any others? Infinitely many? This isn't in the OEIS (at least that I can find).
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I wrote something that can calculate for larger numbers, and have not found any others for any N < 2000.
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Update: no other solutions for N <= 3300. I stopped there as it is taking about 12 seconds to check each further number.
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I would say this now constitutes an interesting open question in elementary number theory -- whether another such N exists ... such questions are hard to come by.
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GM
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October 14th, 2021 at 3:32:27 PM permalink
Quote: teliot

The question about N! not being a perfect square for N >= 2 leads to a question that I don't know the answer to. Here goes.

For any integer N, consider the list of all primes that are in the half-open interval (N/2,N]. Call them p1, p2, p3 ... , pk. The question is, for which N is N!/(p1*p2*p3*...*pk) a perfect square? In other words, divide N! by all the primes greater than N/2 and less than or equal to N (the Bertrand's Postulate primes) and ask if what's leftover is a perfect square.

Here are the five examples I found searching N <= 22 (the largest N for which Excel accurately computes N!).

With N = 6, then we divide 6! by 5 and 6!/5 = 144 is a perfect square.
With N = 7, then we divide 7! by 5*7 = 35 and get 7!/35 = 144, again a perfect square.
With N = 10, then we divide 10! by 7 (the only prime in the interval (5,10]) and get 10!/7 = 518400 = 720^2 is a perfect square.
With N = 11, then we divide 11! by 11*7 and again get 10!/7 = 518400 = 720^2.
With N = 20, then we divide 20! by 19*17*13*11 and get 52672757760000 = 7257600^2.

Any others? Infinitely many? This isn't in the OEIS (at least that I can find).
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There is a paper by Jitsuro Nagura, which proves that for any real number x≥9, there exists a prime p such that x≤p<4x/3. If N≥35, we can apply this theorem with x=(N+1)/4 to obtain that there exists a prime p such that (N+1)/4≤p<(N+1)/3. Then p>N/4 clearly, and 3p<N+1, so 3p≤N, because 3p and N are integers, therefore p≤N/3.The only numbers among 1,2,...,N which are divisible by p are p, 2p and 3p, so N! is divisible by p^3 but not by p^4. After factoring out the primes between N and 2N, the resulting number will still be divisible by p^3, but not by p^4, so it cannot be a square.

This means that if N! divided by the product of primes in (N/2,N] is a square then N≤34. It is easy to check that the only solutions are N=2,3,6,7,10,11,20.
teliot
teliot
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October 14th, 2021 at 6:11:50 PM permalink
Quote: GM

Quote: teliot

The question about N! not being a perfect square for N >= 2 leads to a question that I don't know the answer to. Here goes.

For any integer N, consider the list of all primes that are in the half-open interval (N/2,N]. Call them p1, p2, p3 ... , pk. The question is, for which N is N!/(p1*p2*p3*...*pk) a perfect square? In other words, divide N! by all the primes greater than N/2 and less than or equal to N (the Bertrand's Postulate primes) and ask if what's leftover is a perfect square.

Here are the five examples I found searching N <= 22 (the largest N for which Excel accurately computes N!).

With N = 6, then we divide 6! by 5 and 6!/5 = 144 is a perfect square.
With N = 7, then we divide 7! by 5*7 = 35 and get 7!/35 = 144, again a perfect square.
With N = 10, then we divide 10! by 7 (the only prime in the interval (5,10]) and get 10!/7 = 518400 = 720^2 is a perfect square.
With N = 11, then we divide 11! by 11*7 and again get 10!/7 = 518400 = 720^2.
With N = 20, then we divide 20! by 19*17*13*11 and get 52672757760000 = 7257600^2.

Any others? Infinitely many? This isn't in the OEIS (at least that I can find).
link to original post


There is a paper by Jitsuro Nagura, which proves that for any real number x≥9, there exists a prime p such that x≤p<4x/3. If N≥35, we can apply this theorem with x=(N+1)/4 to obtain that there exists a prime p such that (N+1)/4≤p<(N+1)/3. Then p>N/4 clearly, and 3p<N+1, so 3p≤N, because 3p and N are integers, therefore p≤N/3.The only numbers among 1,2,...,N which are divisible by p are p, 2p and 3p, so N! is divisible by p^3 but not by p^4. After factoring out the primes between N and 2N, the resulting number will still be divisible by p^3, but not by p^4, so it cannot be a square.

This means that if N! divided by the product of primes in (N/2,N] is a square then N≤34. It is easy to check that the only solutions are N=2,3,6,7,10,11,20.
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An elegant and simple argument. Many thanks!

This reminds me of a theorem I have long since forgotten, an improvement on Bertrand's Postulate that follows from the PNT, that for any e > 0 there is a positive integer N such that for all n > N, there exists a prime p with n <= p < (1+e)n. The result you quoted was this bound N being explicitly computed when e equals 1/3.

Again fabulous work.

By the way, I tried to look up your reference, Wikipedia pointed to this result by Nagura: there is always a prime between n and 6/5*n for n >= 25. I couldn't find the result in the paper you referenced. But your proof holds, we would just have to run the exhaustive search out a bit further.
Last edited by: teliot on Oct 14, 2021
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ben771williams
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October 15th, 2021 at 5:18:25 AM permalink
here's another puzzle that's worthy of a beer
GM
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teliot
October 15th, 2021 at 5:45:00 AM permalink
Quote: teliot


By the way, I tried to look up your reference, Wikipedia pointed to this result by Nagura: there is always a prime between n and 6/5*n for n >= 25. I couldn't find the result in the paper you referenced. But your proof holds, we would just have to run the exhaustive search out a bit further.
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There is a link on that page to download the whole paper, the statement of the theorem is at the bottom of the penultimate page.
teliot
teliot
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October 15th, 2021 at 5:50:58 AM permalink
Quote: GM

Quote: teliot


By the way, I tried to look up your reference, Wikipedia pointed to this result by Nagura: there is always a prime between n and 6/5*n for n >= 25. I couldn't find the result in the paper you referenced. But your proof holds, we would just have to run the exhaustive search out a bit further.
link to original post


There is a link on that page to download the whole paper, the statement of the theorem is at the bottom of the penultimate page.
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Okay -- found the download button and the theorem. For reference:



By the way, one consequence of an easy generalization of your argument is that for any integer k there exists an integer N such that for n > N there is a prime p so that p^k exactly divides n!.
Last edited by: teliot on Oct 15, 2021
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Gialmere
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October 18th, 2021 at 7:48:42 AM permalink


Consider a pyramid with a base of a regular N-gon. All other sides are triangles and every edge is of unit length.

1) What is the maximum N?

2) What is the angle between the base and a side?

3) What is the angle between two adjacent sides?
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aceside
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October 18th, 2021 at 5:59:24 PM permalink
Quote: Gialmere



Consider a pyramid with a base of a regular N-gon. All other sides are triangles and every edge is of unit length.

1) What is the maximum N?

2) What is the angle between the base and a side?

3) What is the angle between two adjacent sides?

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Apparently N=5, because when N=6, that pyramid will be flat.
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October 18th, 2021 at 6:56:26 PM permalink

1. 5. At six, the pyramid would be flat. I'm not sure if that counts as a pyramid.
2. I'll bother to answer that if you confirm my answer of 5.
3. Based on a pentagonal base, 108.

I have a feeling I am not understanding the question correctly.
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ChesterDog
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October 18th, 2021 at 7:09:47 PM permalink
Quote: Gialmere



Consider a pyramid with a base of a regular N-gon. All other sides are triangles and every edge is of unit length.

...
2) What is the angle between the base and a side?

...

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