## Sets of 24 consecutive squares whose sum is a square

Sets of 24 consecutive squares beginning with   $m^2$   whose sum is a square,
meaning   $k = 24$

$m^2 \; + \; (m+1)^2 \; + \; (m+2)^2 \; + \; \dotsb \; + \; (m+k-1)^2$
$= \; 1/6 \; (2 \, k^3 + 6 \, k^2 \, m - 3 \, k^2 + 6 \, k \, m^2 - 6 \, k \, m + k)$
$= \; (k/6) \; (2 \, k^2 + 6 \, k \, m - 3 \, k + 6 \, m^2 - 6 \, m + 1)$

There are six infinite families of solutions whose smallest members are :

$1^2 + 2^2 + 3^2 + \dotsb + 24^2 = 70^2$
$121^2 + 122^2 + 123^2 + \dotsb + 144^2 = 650^2$
$1301^2 + 1302^2 + 1303^2 + \dotsb + 1324^2 = 6430^2$
$12981^2 + 12982^2 + 12983^2 + \dotsb + 13004^2 = 63650^2$
$128601^2 + 128602^2 + 128603^2 + \dotsb + 128624^2 = 630070^2$
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$9^2 + 10^2 + 11^2 + \dotsb + 32^2 = 106^2$
$197^2 + 198^2 + 199^2 + \dotsb + 220^2 = 1022^2$
$2053^2 + 2054^2 + 2055^2 + \dotsb + 2076^2 = 10114^2$
$20425^2 + 20426^2 + 20427^2 + \dotsb + 20448^2 = 100118^2$
$202289^2 + 202290^2 + 202291^2 + \dotsb + 202312^2 = 991066^2$
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$20^2 + 21^2 + 22^2 + \dotsb + 43^2 = 158^2$
$304^2 + 305^2 + 306^2 + \dotsb + 327^2 = 1546^2$
$3112^2 + 3113^2 + 3114^2 + \dotsb + 3135^2 = 15302^2$
$30908^2 + 30909^2 + 30910^2 + \dotsb + 30931^2 = 151474^2$
$306060^2 + 306061^2 + 306062^2 + \dotsb + 306083^2 = 1499438^2$
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$25^2 + 26^2 + 27^2 + \dotsb + 48^2 = 182^2$
$353^2 + 354^2 + 355^2 + \dotsb + 376^2 = 1786^2$
$3597^2 + 3598^2 + 3599^2 + \dotsb + 3620^2 = 17678^2$
$35709^2 + 35710^2 + 35711^2 + \dotsb + 35732^2 = 174994^2$
$353585^2 + 353586^2 + 353587^2 + \dotsb + 353608^2 = 1732262^2$
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$44^2 + 45^2 + 46^2 + \dotsb + 67^2 = 274^2$
$540^2 + 541^2 + 542^2 + \dotsb + 563^2 = 2702^2$
$5448^2 + 5449^2 + 5450^2 + \dotsb + 5471^2 = 26746^2$
$54032^2 + 54033^2 + 54034^2 + \dotsb + 54055^2 = 264758^2$
$534964^2 + 534965^2 + 534966^2 + \dotsb + 534987^2 = 2620834^2$
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$76^2 + 77^2 + 78^2 + \dotsb + 99^2 = 430^2$
$856^2 + 857^2 + 858^2 + \dotsb + 879^2 = 4250^2$
$8576^2 + 8577^2 + 8578^2 + \dotsb + 8599^2 = 42070^2$
$84996^2 + 84997^2 + 84998^2 + \dotsb + 85019^2 = 416450^2$
$841476^2 + 841477^2 + 841478^2 + \dotsb + 841499^2 = 4122430^2$
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