26

Does any row of Pascal's triangle contain a Pythagorean triple?

That is,

Are there integers $n,a,b,c$, where $0\leq a,b,c \leq n$, such that $\binom{n}{a}^2+\binom{n}{b}^2=\binom{n}{c}^2$ ?

Using $\binom{n}{r}=\frac{n!}{r!(n-r)!}$, this is equivalent to

$$[b!(n-b)!c!(n-c)!]^2+[a!(n-a)!c!(n-c)!]^2-[a!(n-a)!b!(n-b)!]^2=0$$

I used Excel to check if any of the first 141 rows contains consecutive entries $a,b,c$ such that $\binom{n}{a}^2+\binom{n}{b}^2=\binom{n}{c}^2$, and I did not find any. (Edit: Excel has limited precision, so it is unreliable for this question.) My question does not require that $a,b,c$ are consecutive entries in a row, but I do not know how to check all combinations of three numbers in each row.

Context: I sometimes ask questions about Pascal's triangle, such as this currently unresolved question about three consecutive integers.

Dan
  • 35,053
  • You can also write it as (a!(n-a)!)^(-2) + (b!(n-b)!)^(-2) = (c! (n-c)!)^(-2). So maybe the parametrization https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem#n_=_%E2%88%922 is useful. – user146125 Sep 11 '24 at 14:02
  • 2
    FWIW assuming without loss of generality that $0<a<b<c\leq\tfrac n2$ it follows that $$\binom ba^2+\binom{n-a}{n-b}^2=\left(\binom nc\frac{\binom ba}{\binom na}\right)^2.$$ – Servaes Sep 11 '24 at 18:33
  • 1
    The answer is yes; one example has been found. I have asked a follow-up question at MO: How many other Pythagorean triples are contained in a single row of Pascal's triangle? – Dan Sep 12 '24 at 04:31
  • Another follow-up question: "How many primitive Pythagorean triples are found in the interior of Pascal's triangle?" – Dan Sep 18 '24 at 05:59

1 Answers1

35

Check:

$$\binom{62}{26}^2+ \binom{62}{27}^2 = \binom{62}{28}^2 $$

I can share my Mathematica code if you want so. I am still running the program but so far this is the only solution.

Oldboy
  • 17,264
  • 11
    I checked it by hand, and if you remove the shared factors it boils down to $3^2 + 4^2 = 5^2$. Really cool. – user146125 Sep 11 '24 at 15:52
  • 1
    I have asked a follow-up question at MO: How many other Pythagorean triples are contained in a single row of Pascal's triangle? – Dan Sep 12 '24 at 04:32
  • 1
    Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on [meta], or in [chat]. Comments continuing discussion may be removed. – Shaun Sep 12 '24 at 17:11