1 + 2 + 3 + … + m^2 = A^2

 
 
Let   S(n)   denote the sum of the first   n   positive integers where both   n   and   S(n)   are perfect squares.

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; m^2 \; = \; A^2,     where    n = m^2
 
 
Here are the first few solutions:
 

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 7^2
= \; 1225
= \; 35^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 41^2
= \; 1413721
= \; 1189^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 239^2
= \; 1631432881
= \; 40391^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 1393^2
= \; 1882672131025
= \; 1372105^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 8119^2
= \; 2172602007770041
= \; 46611179^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 47321^2
= \; 2507180834294496361
= \; 1583407981^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 275807^2
= \; 2893284510173841030625
= \; 53789260175^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 1607521^2
= \; 3338847817559778254844961
= \; 1827251437969^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 9369319^2
= \; 3853027488179473932250054441
= \; 62072759630771^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 54608393^2
= \; 4446390382511295358038307980025
= \; 2108646576008245^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 318281039^2
= \; 5131130648390546663702275158894481
= \; 71631910824649559^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 1855077841^2
= \; 5921320321852308338617067495056251121
= \; 2433376321462076761^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 10812186007^2
= \; 6833198520286915432217432187019754899225
= \; 82663163018885960315^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 63018038201^2
= \; 7885505171090778556470578126753302097454601
= \; 2808114166320660573949^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 367296043199^2
= \; 9099866134240238167251614940841123600707710401
= \; 95393218491883573553951^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 2140758220993^2
= \; 10501237633408063754229807171152529881914600348225
= \; 3240561314557720840260385^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 12477253282759^2
= \; 12118419129086771332143030223895078642605848094141321
= \; 110083691476470624995299139^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 72722761475561^2
= \; 13984645173728500709229302648567749601037266786038736281
= \; 3739604948885443528999910341^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 423859315570607^2
= \; 16138268412063560731679283113416959144518363265240607527025
= \; 127036484570628609361001652455^2

1 \; + \; 2 \; + \; 3 \; + \; ... \; + \; 2470433131948081^2
= \; 18623547762876175355857183483580522285024590170820875047450641
= \; 4315500870452487274745056273129^2

………………………………………………….
………………………………………………….

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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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