n and n+1 consecutive integers with equal sums of squares –(Part 1)

 
 
n = 2

13^2 + 14^2 \; = \; 10^2 + 11^2 + 12^2

133^2 + 134^2 \; = \; 108^2 + 109^2 + 110^2

1321^2 + 1322^2 \; = \; 1078^2 + 1079^2 + 1080^2

13081^2 + 13082^2 \; = \; 10680^2 + 10681^2 + 10682^2

129493^2 + 129494^2 \; = \; 105730^2 + 105731^2 + 105732^2

1281853^2 + 1281854^2 \; = \; 1046628^2 + 1046629^2 + 1046630^2

12689041^2 + 12689042^2 \; = \; 10360558^2 + 10360559^2 + 10360560^2

125608561^2 + 125608562^2 \; = \; 102558960^2 + 102558961^2 + 102558962^2

1243396573^2 + 1243396574^2 \; = \; 1015229050^2 + 1015229051^2 + 1015229052^2

 

n = 3

25^2 + 26^2 + 27^2 \; = \; 21^2 + 22^2 + 23^2 + 24^2

361^2 + 362^2 + 363^2 \; = \; 312^2 + 313^2 + 314^2 + 315^2

5041^2 + 5042^2 + 5043^2 \; = \; 4365^2 + 4366^2 + 4367^2 + 4368^2

70225^2 + 70226^2 + 70227^2 \; = \; 60816^2 + 60817^2 + 60818^2 + 60819^2

978121^2 + 978122^2 + 978123^2
= \; 847077^2 + 847078^2 + 847079^2 + 847080^2

13623481^2 + 13623482^2 + 13623483^2
= \; 11798280^2 + 11798281^2 + 11798282^2 + 11798283^2

189750625^2 + 189750626^2 + 189750627^2
= \; 164328861^2 + 164328862^2 + 164328863^2 + 164328864^2

2642885281^2 + 2642885282^2 + 2642885283^2
= \; 2288805792^2 + 2288805793^2 + 2288805794^2 + 2288805795^2

36810643321^2 + 36810643322^2 + 36810643323^2
= \; 31878952245^2 + 31878952246^2 + 31878952247^2 + 31878952248^2

 
n = 4

41^2 + 42^2 + 43^2 + 44^2 \; = \; 36^2 + 37^2 + 38^2 + 39^2 + 40^2

761^2 + 762^2 + 763^2 + 764^2 \; = \; 680^2 + 681^2 + 682^2 + 683^2 + 684^2

13681^2 + 13682^2 + 13683^2 + 13684^2
= \; 12236^2 + 12237^2 + 12238^2 + 12239^2 + 12240^2

245521^2 + 245522^2 + 245523^2 + 245524^2
= \; 219600^2 + 219601^2 + 219602^2 + 219603^2 + 219604^2

4405721^2 + 4405722^2 + 4405723^2 + 4405724^2
= \; 3940596^2 + 3940597^2 + 3940598^2 + 3940599^2 + 3940600^2

79057481^2 + 79057482^2 + 79057483^2 + 79057484^2
= \; 70711160^2 + 70711161^2 + 70711162^2 + 70711163^2 + 70711164^2

1418628961^2 + 1418628962^2 + 1418628963^2 + 1418628964^2
= \; 1268860316^2 + 1268860317^2 + 1268860318^2 + 1268860319^2 + 1268860320^2

25456263841^2 + 25456263842^2 + 25456263843^2 + 25456263844^2
= \; 22768774560^2 + 22768774561^2 + 22768774562^2 + 22768774563^2 + 22768774564^2

456794120201^2 + 456794120202^2 + 456794120203^2 + 456794120204^2
= \; 408569081796^2 + 408569081797^2 + 408569081798^2 + 408569081799^2 + 408569081800^2

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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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