Can we Mordell it up?

Number Theory Level 5

Except for the trivial solution of $(m,n) = (1,1),$ how many other pairs of positive integers $(m,n)$ satisfy

$1 + 2 + 3 + \cdots + m = 1 + (1+2) + ( 1 +2 + 3) + \cdots + (1 + 2 + 3 + \cdots + n) ?$

1 2 3 4 Infinitely many

1 solution

Mohammad Farhat
Oct 15, 2018

Relevant wiki: Gauss: The Prince of Mathematics

It has to satisfy the equation:

$T_n = \dbinom{n+1}{2} = \dbinom{m+2}{3} = Te_m$

The only solutions of which are :

$Te_3 = T_4 =10$

$Te_8 = T_{15} =120$

$Te_{20} = T_{55} = 1540$

$Te_{34} = T_{119} = 7140$

Your solution is incomplete. You didn't justify why there is only 1 solution.

Pi Han Goh - 2 years, 7 months ago

@Pi Han Goh , I made a report.

Mohammad Farhat - 2 years, 7 months ago

@Pi Han Goh , There is more than 1 solution.

Mohammad Farhat - 2 years, 7 months ago

@Calvin Lin , Can you please add an option (that is 4) and make it the correct answer.

Mohammad Farhat - 2 years, 7 months ago

@Calvin Lin , I think that it is rather queer that I got the question wrong while I am able to post a solution. Can you do something about it?

Mohammad Farhat - 2 years, 7 months ago

You were marked correct at the start because you got the (now incorrect) "correct answer" of 1. That is what allowed you to submit a solution (before the answers were changed).

Typically, if someone was marked incorrect, then they would be unable to submit a solution.

Calvin Lin Staff - 2 years, 7 months ago

Can we Mordell it up?

1 solution

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