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

**n = 5**

**n = 6**

**n = 7**

**n = 8**

**n = 9**

**n = 10**

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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.

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.

I did the following:

For n = 7

yet, I couldn’t find solutions!

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.

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]

Thanks!