Integer with 2 representations as a sum of 2 positive 4-th powers

That is to say,   N   =   a^4   +   b^4   =   c^4   +   d^4
where N, a, b, c, and d are all different, non-zero, positive integers
 
This is a problem that calls for a brute-force program.
I posted below only 12 results.
 
(1)

635318657   =   59^4   +   158^4   =   133^4   +   134^4

Interestingly,
635318657   has 8 representations as a sum of 2 squares:
635318657   =   3481^2   +   24964^2   =   6764^2   +   24281^2   =
8876^2   +   23591^2   =   11929^2   +   22204^2   =   13316^2   +   21401^2 =
15449^2   +   19916^2   =   16039^2   +   19444^2   =   17689^2   +   17956^2
 
 
(2)

3262811042   =   7^4   +   239^4   =   157^4   +   227^4

And,
3262811042   has 4 representations as a sum of 2 squares:
3262811042   =   49^2   +   57121^2   =   7631^2   +   56609^2   =   24649^2   +   51529^2   =   31271^2   +   47801^2
 
 
(3)

8657437697   =   193^4   +   292^4   =   256^4   +   257^4
(256,   257   are consecutive integers)

And,
8657437697   has 8 representations as a sum of 2 squares:
8657437697 = 6529^2 + 92816^2 = 17624^2 + 91361^2 = 26744^2 + 89119^2 =
37249^2 + 85264^2 = 49439^2 + 78824^2 = 57704^2 + 72991^2 =
58544^2 + 72319^2 = 65536^2 + 66049^2
 
 
(4)

10165098512   =   118^4   +   316^4   =   266^4   +   268^4
(266,   268 are consecutive even numbers)
 
 
(5)

51460811217   =   177^4   +   474^4   =   399^4 +   402^4
 
 

(6)

52204976672   =   14^4   +   478^4   =   314^4   +   454^4
 
 
(7)

68899596497   =   271^4 +   502^4   =   298^4   +   497^4
 
 
(8)

86409838577   =   103^4   +   542^4   =   359^4   +   514^4
 
 
(9)

138519003152   =   386^4   +   584^4   =   512^4   +   514^4
(512,   514 consecutive even numbers)
 
 
(10)

160961094577   =   222^4   +   631^4   =   503^4   +   558^4
 
 
(11)

162641576192   =   236^4   +   632^4   =   532^4   +   536^4
(632   is the digit reversal of   236)
 
 
(12)

264287694402   =   21^4   +   717^4   =   471^4   +   681^4
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Math Beauty, Number Puzzles and tagged , . Bookmark the permalink.

1 Response to Integer with 2 representations as a sum of 2 positive 4-th powers

  1. Pingback: Integer with 2 representations as a sum of 2 positive 4-th powers | Benvitalis's Blog

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