Determine the only possible integer value of N, such that each of:
8N3-20, and 2N5-2
is a perfect square.
Prove that no further value of N satisfies the given conditions.
**** Computer program/ excel solver assisted solutions are welcome, but a semi-analytic - namely, p&p and hand calculator based methodology is preferred.
Start with 8n^3-20=a^2, or (2n)^3=a^2+20.
A list of closest powers (prepared for 'Checking the quantity') gives only one pair whose difference is 20: 6^3-20=196=14^2
Since 6=2n, this implies that n is 3. It is a given that the solution (whatever it is) is unique. So if 2(3)^5-2 is also square, then we are done: 2(3)^5-2=484=22^2.
So n=3, and the list of closest powers rules out any other solution.
Edited on September 24, 2023, 11:03 pm
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Posted by broll
on 2023-09-24 23:00:53 |
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