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Quarting the cube (Posted on 2013-02-28) Difficulty: 3 of 5

Select integer x and triangular number y such that 8y=3x^4-2x^2-1.

Prove that y is divisible by 28 - or find a counter-example.

See The Solution Submitted by broll    
Rating: 5.0000 (2 votes)

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re: SOLUTION +some add-ons | Comment 5 of 6 |
(In reply to SOLUTION +some add-ons by Ady TZIDON)

My esteemed Ady,

BTW: all the triangle numbers related to this problem are 4/ 3 of a valid triangle number.

Naturally, since that is implicit in the original problem. Assuming the workings from my solution, to save space:

Let 3N+1 = a^2, and let N=n^2-n 
Let a=(2p-1)
3n^2-3n+1 = 4p^2-4p+1
Cancelling 1 from both sides:
4p^2-4p=3n^2-3n [2]; 4 times a triangular number is 3 times another triangular number.

Checking: a={1,13,181,2521,...}, b={1,15,209,2911,...}, n={1,8,105,1456,...}, p=1,7,91,1261,...},  x={1,3,11,41,..}, y={0,56,10920,2118480,...}. From which we can see at once, for example, that 4*91*90=3*105*104, etc., etc. Hence the statements 8y=3x^4-2x^2-1 and 4p^2-4p=3n^2-3n are essentially equivalent.

 


  Posted by broll on 2013-03-02 06:22:23
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