Positive integers (x,y) such that (x+1)^3 – x^3 = y^2

 
 

x   and   y   are positive integers that satisfy the equation

(x+1)^3 \; - \; x^3 \; = \; y^2

 

Here are the first few solutions:

x =   7,   104,   1455,   20272,   282359,   3932760,   54776287,   762935264,   10626317415
 
 

(x+1)^3 \; - \; x^3 \; = \; y^2 ……………………………………….   y \; = \; a^2 \; + \; (a+1)^2

8^3 - 7^3 = 13^2 ……………………………………………………. 13 = 2^2 + 3^2
105^3 - 104^3 = 181^2 ……………………………………………. 181 = 9^2 + 10^2
1456^3 - 1455^3 = 2521^2 ………………………………………. 2521 = 35^2 + 36^2
20273^3 - 20272^3 = 35113^2 …………………………………. 35113 = 132^2 + 133^2
282360^3 - 282359^3 = 489061^2 ……………………………. 489061 = 494^2 + 495^2
3932761^3 - 3932760^3 =  6811741^2 ………………………. 6811741 = 1845^2 + 1846^2
54776288^3 - 54776287^3 = 94875313^2 ………………….. 94875313 = 6887^2 + 6888^2
762935265^3 - 762935264^3 = 1321442641^2 …………… 1321442641 = 25704^2 + 25705^2
10626317416^3 - 10626317415^3 = 18405321661^2 ….. 18405321661 = 95930^2 + 95931^2

 

Prove that y is always expressible as the sum of squares of two consecutive positive integers.

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

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