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.)

computer confirmation

| Comment 2 of 9 |

fwiw, a computer run up to n=10,000 finds only the same 3 solutions found by K Sengupta:

n r

1 9.0

2 20.0

5 55.0

Note that s3(1) = 1^3 + 2^3 = 9 rather than 1

--------

def s3(n):

ans = 0

for i in range(1,2*n+1):

ans += i**3

return ans

def s2(n):

ans = 0

for i in range(1,n+1):

ans += i**2

return ans

for n in range(1,10001):

r = (s3(n)/s2(n))

if r%1 == 0:

print(n,r)

Posted by Larry on 2022-05-19 09:23:26

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