perplexus.info :: Just Math : Only integer answers

Only integer answers (Posted on 2022-05-19)

Let

S(3)= 1³+2³+…+(2n)³
and
S(2)= 1²+2²+…+n²

For what integer values of n will the ratio r=S(3)/S(2) have an integer result?

List all existing answers.

Source: Russian Math Olympiad

See The Solution Submitted by Ady TZIDON

Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

Matlab version

| Comment 4 of 9 |

(In reply to computer confirmation by Larry)

clearvars , clc

s3=sym(0); s2=sym(0);

for n=1:9999999

s3term=sym(2*n-1)^3+sym(2*n)^3; s2term=sym(n)^2;

s3=s3+s3term; s2=s2+s2term;

if s3/s2==round(s3/s2)

disp([n s3 s2 s3/s2])

end

finds

[1, 9, 1, 9]

[2, 100, 5, 20]

[5, 3025, 55, 55]

showing n, s3, s2, s3/s2 for each.

It was actually stopped around n = 20,000.

Posted by Charlie on 2022-05-19 10:21:07

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