#1
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In the spirit of jay_shark
I forget if I've posted this one before, but I like it because it's a problem I posed to myself a few years ago:
Can every positive integer be represented as the ratio of two triangle numbers? |
#2
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Re: In the spirit of jay_shark
LOL, I first read this as "can ANY ..."
Not sure if it's helpfull, but this is equivalent to asking: can all positive integers be represented by: n(n+1)/[m(m+1)] where n and m are positive integers? |
#3
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Re: In the spirit of jay_shark
This is very interesting Gump .
I think you may be right . I'll think about this one . |
#4
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Re: In the spirit of jay_shark
It's been a couple days, so I think I can post a solution without stomping on anyone's toes....
The number 4 cannot be so represented. if it could be, we would have 4n(n+1)=m(m+1) for some integers m,n. But m=2n is too small, and m=2n+1 is too large. 4n^2+2n < 4n^2 + 4n < 4n^2 + 6n + 2 for all positive n. Same proof works for 9. Looks like something just slightly different will work for any perfect square. |
#5
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Re: In the spirit of jay_shark
[ QUOTE ]
It's been a couple days, so I think I can post a solution without stomping on anyone's toes.... The number 4 cannot be so represented. if it could be, we would have 4n(n+1)=m(m+1) for some integers m,n. But m=2n is too small, and m=2n+1 is too large. 4n^2+2n < 4n^2 + 4n < 4n^2 + 6n + 2 for all positive n. Same proof works for 9. Looks like something just slightly different will work for any perfect square. [/ QUOTE ] Prime factorizations lead somewhere too. |
#6
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Re: In the spirit of jay_shark
[ QUOTE ]
It's been a couple days, so I think I can post a solution without stomping on anyone's toes.... The number 4 cannot be so represented. if it could be, we would have 4n(n+1)=m(m+1) for some integers m,n. But m=2n is too small, and m=2n+1 is too large. 4n^2+2n < 4n^2 + 4n < 4n^2 + 6n + 2 for all positive n. Same proof works for 9. Looks like something just slightly different will work for any perfect square. [/ QUOTE ] Yep, at least on the 4. The perfect square argument sounds pretty plausible to me, too. |
#7
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Re: In the spirit of jay_shark
The perfect square argument runs onto the rocks for 36/1 and 300/3.
I misspoke - only squares of prime numbers. (The basic idea is that k^2 n (n+1) can't be rearranged to two consecutive numbers near kn since one is divisible by k and the other isn't. But for 6^2 or 10^2, it's possible to find consecutive numbers where one is a multiple of 4 and other a multiple of 9 or 25 -- 8*9/2 and 24*25/2.) Has anyone been able to prove that it IS possible for all OTHER numbers? |
#8
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Re: In the spirit of jay_shark
[ QUOTE ]
The perfect square argument runs onto the rocks for 36/1 and 300/3. I misspoke - only squares of prime numbers. (The basic idea is that k^2 n (n+1) can't be rearranged to two consecutive numbers near kn since one is divisible by k and the other isn't. But for 6^2 or 10^2, it's possible to find consecutive numbers where one is a multiple of 4 and other a multiple of 9 or 25 -- 8*9/2 and 24*25/2.) Has anyone been able to prove that it IS possible for all OTHER numbers? [/ QUOTE ] No prime powers other than 2 or 3. Consider prime factorizations of all numbers involved. |
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