## n and n+1 consecutive integers with equal sums of squares –(Part 2)

n = 5

$61^2 + 62^2 + 63^2 + 64^2 + 65^2$
$= 55^2 + 56^2 + 57^2 + 58^2 + 59^2 + 60^2$

$1381^2 + 1382^2 + 1383^2 + 1384^2 + 1385^2$
$= 1260^2 + 1261^2 + 1262^2 + 1263^2 + 1264^2 + 1265^2$

$30361^2 + 30362^2 + 30363^2 + ... + 30365^2$
$= 27715^2 + 27716^2 + 27717^2 + ... + 27720^2$

$666601^2 + 666602^2 + 666603^2 + ... + 666605^2$
$= 608520^2 + 608521^2 + 608522^2 + ... + 608525^2$

$14634901^2 + 14634902^2 + 14634903^2 + ... + 14634905^2$
$= 13359775^2 + 13359776^2 + 13359777^2 + ... + 13359780^2$

$321301261^2 + 321301262^2 + 321301263^2 + ... + 321301265^2$
$= 293306580^2 + 293306581^2 + 293306582^2 + ... + 293306585^2$

$7053992881^2 + 7053992882^2 + 7053992883^2 + ... + 7053992885^2$
$= 6439385035^2 + 6439385036^2 + 6439385037^2 + ... + 6439385040^2$

$154866542161^2 + 154866542162^2 + 154866542163^2 + ... + 154866542165^2$
$= 141373164240^2 + 141373164241^2 + 141373164242^2 + ... + 141373164245^2$

$3400009934701^2 + 3400009934702^2 + 3400009934703^2 + ... + 3400009934705^2$
$= 3103770228295^2 + 3103770228296^2 + 3103770228297^2 + ... + 3103770228300^2$

$74645352021301^2 + 74645352021302^2 + 74645352021303^2 + ... + 74645352021305^2$
$= 68141571858300^2 + 68141571858301^2 + 68141571858302^2 + ... + 68141571858305^2$

n = 6

$85^2 + 86^2 + 87^2 + 88^2 + 89^2 + 90^2$
$= 78^2 + 79^2 + 80^2 + 81^2 + 82^2 + 83^2 + 84^2$

$2269^2 + 2270^2 + 2271^2 + 2272^2 + 2273^2 + 2274^2$
$= 2100^2 + 2101^2 + 2102^2 + 2103^2 + 2104^2 + 2105^2 + 2106^2$

$58969^2 + 58970^2 + 58971^2 + ... + 58974^2$
$= 54594^2 + 54595^2 + 54596^2 + ... + 54600^2$

$1530985^2 + 1530986^2 + 1530987^2 + ... + 1530990^2$
$= 1417416^2 + 1417417^2 + 1417418^2 + ... + 1417422^2$

$39746701^2 + 39746702^2 + 39746703^2 + ... + 39746706^2$
$= 36798294^2 + 36798295^2 + 36798296^2 + ... + 36798300^2$

$1031883301^2 + 1031883302^2 + 1031883303^2 + ... + 1031883306^2$
$= 955338300^2 + 955338301^2 + 955338302^2 + ... + 955338306^2$

$26789219185^2 + 26789219186^2 + 26789219187^2 + ... + 26789219190^2$
$= 24801997578^2 + 24801997579^2 + 24801997580^2 + ... + 24801997584^2$

$695487815569^2 + 695487815570^2 + 695487815571^2 + ... + 695487815574^2$
$= 643896598800^2 + 643896598801^2 + 643896598802^2 + ... + 643896598806^2$

$18055893985669^2 + 18055893985670^2 + 18055893985671^2 + ... + 18055893985674^2$
$= 16716509571294^2 + 16716509571295^2 + 16716509571296^2 + ... + 16716509571300^2$

$468757755811885^2 + 468757755811886^2 + 468757755811887^2 + ... + 468757755811890^2$
$= 433985352254916^2 + 433985352254917^2 + 433985352254918^2 + ... + 433985352254922^2$

n = 7

$113^2 + 114^2 + 115^2 + ... + 119^2$
$= 105^2 + 106^2 + 107^2 + ... + 112^2$

n = 8

$145^2 + 146^2 + 147^2 + ... + 152^2$
$= \; 136^2 + 137^2 + 138^2 + ... + 144^2$

$862^2 + 863^2 + 864^2 + ... + 869^2$
$= \; 812^2 + 813^2 + 814^2 + ... + 820^2$

$5041^2 + 5042^2 + 5043^2 + ... + 5048^2$
$= \; 4752^2 + 4753^2 + 4754^2 + ... + 4760^2$

$171361^2 + 171362^2 + 171363^2 + ... + 171368^2$
$= \; 161560^2 + 161561^2 + 161562^2 + ... + 161568^2$

$29398^2 + 29399^2 + 29400^2 + ... + 29405^2$
$= \; 27716^2 + 27717^2 + 27718^2 + ... + 27724^2$

$5821345^2 + 5821346^2 + 5821347^2 + ... + 5821352^2$
$= \; 5488416^2 + 5488417^2 + 5488418^2 + ... + 5488424^2$

$998782^2 + 998783^2 + 998784^2 + ... + 998789^2$
$= \; 941660^2 + 941661^2 + 941662^2 + ... + 941668^2$

$33929302^2 + 33929303^2 + 33929304^2 + ... + 33929309^2$
$= \; 31988852^2 + 31988853^2 + 31988854^2 + ... + 31988860^2$

$197754481^2 + 197754482^2 + 197754483^2 + ... + 197754488^2$
$= \; 186444712^2 + 186444713^2 + 186444714^2 + ... + 186444720^2$

$1152597598^2 + 1152597599^2 + 1152597600^2 + ... + 1152597605^2$
$= \; 1086679436^2 + 1086679437^2 + 1086679438^2 + ... + 1086679444^2$

n = 9

$181^2 + 182^2 + 183^2 + ... + 189^2 = 171^2 + 172^2 + 173^2 + ... + 180^2$

$7021^2 + 7022^2 + 7023^2 + ... + 7029^2$
$= \; 6660^2 + 6661^2 + 6662^2 + ... + 6669^2$

$266761^2 + 266762^2 + 266763^2 + ... + 266769^2$
$= \; 253071^2 + 253072^2 + 253073^2 + ... + 253080^2$

$10130041^2 + 10130042^2 + 10130043^2 + ... + 10130049^2$
$= \; 9610200^2 + 9610201^2 + 9610202^2 + ... + 9610209^2$

$384674941^2 + 384674942^2 + 384674943^2 + ... + 384674949^2$
$= \; 364934691^2 + 364934692^2 + 364934693^2 + ... + 364934700^2$

$14607517861^2 + 14607517862^2 + 14607517863^2 + ... + 14607517869^2$
$= \; 13857908220^2 + 13857908221^2 + 13857908222^2 + ... + 13857908229^2$

$554701003921^2 + 554701003922^2 + 554701003923^2 + ... + 554701003929^2$
$= \; 526235577831^2 + 526235577832^2 + 526235577833^2 + ... + 526235577840^2$

$21064030631281^2 + 21064030631282^2 + 21064030631283^2 + ... + 21064030631289^2$
$= \; 19983094049520^2 + 19983094049521^2 + 19983094049522^2 + ... + 19983094049529^2$

$799878462984901^2 + 799878462984902^2 + 799878462984903^2 + ... + 799878462984909^2$
$= \; 758831338304091^2 + 758831338304092^2 + 758831338304093^2 + ... + 758831338304100^2$

$30374317562795101^2 + 30374317562795102^2 + ... + 30374317562795109^2$
$= 28815607761506100^2 + 28815607761506101^2 + ... + 28815607761506109^2$

n = 10

$221^2 + 222^2 + 223^2 + ... + 230^2$
$= 210^2 + 211^2 + 212^2 + ... + 220^2$

$9461^2 + 9462^2 + 9463^2 + ... + 9470^2$
$= 9020^2 + 9021^2 + 9022^2 + ... + 9030^2$

$397321^2 + 397322^2 + 397323^2 + ... + 397330^2$
$= 378830^2 + 378831^2 + 378832^2 + ... + 378840^2$

$16678201^2 + 16678202^2 + 16678203^2 + ... + 16678210^2$
$= 15902040^2 + 15902041^2 + 15902042^2 + ... + 15902050^2$

$700087301^2 + 700087302^2 + 700087303^2 + ... + 700087310^2$
$= 667507050^2 + 667507051^2 + 667507052^2 + ... + 667507060^2$

$29386988621^2 + 29386988622^2 + 29386988623^2 + ... + 29386988630^2$
$= 28019394260^2 + 28019394261^2 + 28019394262^2 + ... + 28019394270^2$

$1233553434961^2 + 1233553434962^2 + 1233553434963^2 + ... + 1233553434970^2$
$= 1176147052070^2 + 1176147052071^2 + 1176147052072^2 + ... + 1176147052080^2$

$51779857279921^2 + 51779857279922^2 + 51779857279923^2 + ... + 51779857279930^2$
$= 49370156792880^2 + 49370156792881^2 + 49370156792882^2 + ... + 49370156792890^2$

$2173520452321901^2 + 2173520452321902^2 + 2173520452321903^2 + ... + 2173520452321910^2$
$= 2072370438249090^2 + 2072370438249091^2 + 2072370438249092^2 + ... + 2072370438249100^2$

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## About benvitalis

math grad - Interest: Number theory
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### 6 Responses to n and n+1 consecutive integers with equal sums of squares –(Part 2)

1. paul says:

There are solutions for n = 7 and a smaller one for n = 8. { a and c } are the start of the terms

2 {a->13,c->10}
{a->133,c->108}
{a->1321,c->1078}
{a->13081,c->10680}
{a->129493,c->105730}
{a->1281853,c->1046628}
{a->12689041,c->10360558}
{a->125608561,c->102558960}
{a->1243396573,c->1015229050}

3 {a->25,c->21}
{a->361,c->312}
{a->5041,c->4365}
{a->70225,c->60816}
{a->978121,c->847077}
{a->13623481,c->11798280}
{a->189750625,c->164328861}
{a->2642885281,c->2288805792}

4 {a->41,c->36}
{a->761,c->680}
{a->13681,c->12236}
{a->245521,c->219600}
{a->4405721,c->3940596}
{a->79057481,c->70711160}
{a->1418628961,c->1268860316}

5 {a->61,c->55}
{a->1381,c->1260}
{a->30361,c->27715}
{a->666601,c->608520}
{a->14634901,c->13359775}
{a->321301261,c->293306580}
{a->7053992881,c->6439385035}

6 {a->85,c->78}
{a->2269,c->2100}
{a->58969,c->54594}
{a->1530985,c->1417416}
{a->39746701,c->36798294}
{a->1031883301,c->955338300}

7 {a->113,c->105}
{a->3473,c->3248}
{a->104161,c->97433}
{a->3121441,c->2919840}
{a->93539153,c->87497865}
{a->2803053233,c->2622016208}

8 {a->22,c->20}
{a->145,c->136}
{a->862,c->812}
{a->5041,c->4752}
{a->29398,c->27716}
{a->171361,c->161560}
{a->998782,c->941660}
{a->5821345,c->5488416}
{a->33929302,c->31988852}
{a->197754481,c->186444712}
{a->1152597598,c->1086679436}
{a->6717831121,c->6333631920}

9 {a->181,c->171}
{a->7021,c->6660}
{a->266761,c->253071}
{a->10130041,c->9610200}
{a->384674941,c->364934691}

10 {a->221,c->210}
{a->9461,c->9020}
{a->397321,c->378830}
{a->16678201,c->15902040}
{a->700087301,c->667507050}

Paul.

2. paul says:

Here are a few for 11 to 20

11 {a->265,c->253}
{a->12409,c->11880}
{a->570769,c->546469}
{a->26243185,c->25125936}
{a->1206615961,c->1155246829}

12 {a->313,c->300}
{a->15913,c->15288}
{a->795601,c->764388}
{a->39764401,c->38204400}
{a->1987424713,c->1909455900}

13 {a->365,c->351}
{a->20021,c->19292}
{a->1081081,c->1041755}
{a->58358665,c->56235816}
{a->3150287141,c->3035692647}

14 {a->421,c->406}
{a->24781,c->23940}
{a->1437241,c->1388506}
{a->83335561,c->80509800}
{a->4832025661,c->4668180286}

15 {a->481,c->465}
{a->30241,c->29280}
{a->1874881,c->1815345}
{a->116212801,c->112522560}
{a->7203319201,c->6974583825}

16 {a->545,c->528}
{a->36449,c->35360}
{a->2405569,c->2333744}
{a->158731585,c->153992256}

17 {a->613,c->595}
{a->43453,c->42228}
{a->3041641,c->2955943}
{a->212871961,c->206874360}

18 {a->685,c->666}
{a->51301,c->49932}
{a->3796201,c->3694950}
{a->280868185,c->273377016}

19 {a->761,c->741}
{a->60041,c->58520}
{a->4683121,c->4564541}
{a->365224081,c->355976400}

20 {a->841,c->820}
{a->69721,c->68040}
{a->5717041,c->5579260}
{a->468728401,c->457432080}

P.

• benvitalis says:

I did the following:
$x^2 + (x+1)^2 + (x+2)^2 + ... + (x+n-1)^2$
$= n \, x^2 + n(n-1) \, x + n(n-1)(2 \, n-1)/6$

$y^2 + (y+1)^2 + (y+2)^2 + ... + (y+n)^2$
$= (n+1) \, y^2 + n(n+1) \, y + n(n+1) \, (2 \,n+1)/6$

$n \, x^2 + n(n-1) \, x + n(n-1)(2 \, n-1)/6$
$= (n+1) \, y^2 + n(n+1) \, y + n(n+1) \, (2 \,n+1)/6$

$n^2 \, x - n^2 \, y - n^2 + n \, x^2 - n \, x - n \, y^2 - n \, y - y^2 = 0$

For n = 7
$7^2 \, x - 7^2 \, y - 7^2 + 7 \, x^2 - 7 \, x - 7 \, y^2 - 7 \, y - y^2 = 0$

yet, I couldn’t find solutions!

3. paul says:

Here’s my code, using solve

Clear[a, b, c, d]; Do[
Print[m, ” “,
Column[p =
Solve[Expand[Sum[i^2, {i, a, a + m – 1}]] ==
Expand[Sum[i^2, {i, c, c + m}]] && a > 0 && c > 0 &&
a < 10^10 && c < a, {a, c}, Integers]]];
p = {a, c} /. p, {m, 2, 10}][/sourcecode]

try these

91 + 42 a + 7 a^2 = 140 + 56 b + 8 b^2

which come from Expand[Sum[i^2, {i, a, a + 6}]] & Expand[Sum[i^2, {i, b, b + 7}]]

Solve basically solve those two equations

Paul.

• paul says:

Clear[a, b, c, d]; Do[
Print[m, ” “,
Column[p =
Solve[Expand[Sum[i^2, {i, a, a + m – 1}]] ==
Expand[Sum[i^2, {i, c, c + m}]] && a > 0 && c > 0 &&
a < 10^10 && c < a, {a, c}, Integers]]];
p = {a, c} /. p, {m, 2, 10}]
[/sourcecode]