# How come, for any n > 2, the nth triangular number + the nth square

by goldust
Tags: number, square, triangular
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 P: 85 number cannot be prime? I have checked this for n from 3 to 53,509, the latter being the limit for unsigned int. I believe this is true, and I thereby claim that this is a true statement. However, I don't see any obvious explanation for it.
 HW Helper P: 3,515 The nth triangle number Tn is $$T_n = \frac{n(n+1)}{2}$$ and the nth square number Sn is $$S_n = n^2$$ And so $$T_n+S_n = \frac{n(n+1)}{2}+n^2=\frac{n(3n+1)}{2}$$ Can you now show why this expression must be composite (not prime) for all n?
P: 85
 Can you now show why this expression must be composite (not prime) for all n?
Well, it wouldn't be true for all n. Only for n > 2, assuming my statement is correct.

 P: 85 How come, for any n > 2, the nth triangular number + the nth square When n is odd and more than 2, 1 / 2 * (3n + 1) is a whole number bigger than 1, and so the result is composite. When n is even and more than 2, n * (3n + 1) is even, so (n / 2) * (3n + 1) is even. Therefore the sum is composite for all n > 2 is correct. Many thanks for the help.
HW Helper
P: 3,515
 Quote by goldust Well, it wouldn't be true for all n. Only for n > 2, assuming my statement is correct.
Well, ignoring the fact that you included the criteria that n>2 in your proof below, $n\leq 2$ would also spit out composite numbers, right? Your proof only considers that n is even or odd which means that all integers $n\leq 2$ would also be involved. The only reason to restrict yourself to n>2 is such that we have a meaningful square and triangle number.

 Quote by goldust When n is odd and more than 2, 1 / 2 * (3n + 1) is a whole number bigger than 1, and so the result is composite.
Adding to the end of that: because we then have a product of two integers, mainly n and $\frac{3n+1}{2}$.

 Quote by goldust When n is even and more than 2, n * (3n + 1) is even, so (n / 2) * (3n + 1) is even. Therefore the sum is composite for all n > 2 is correct. Many thanks for the help.
This is incorrect. If $n(3n+1)$ is even, then $\frac{n}{2}(3n+1)$ isn't necessarily even, but rather an integer. But most importantly, you haven't proven that the expression is a product of two integers and hence composite.

Also, while it's not absolutely necessary, when you consider n to be even, you could let n=2k for some integer k and substitute that into your expression, then show that the result is composite, and similarly for n odd, let n=2k+1.
 P: 85 Many thanks.
P: 1,395
 Quote by Mentallic Well, ignoring the fact that you included the criteria that n>2 in your proof below, $n\leq 2$ would also spit out composite numbers, right? Your proof only considers that n is even or odd which means that all integers $n\leq 2$ would also be involved. The only reason to restrict yourself to n>2 is such that we have a meaningful square and triangle number.
You do get a prime number for n=1 or n=2, and the proof also uses the fact that n>2, when it says
 When n is odd and more than 2, 1 / 2 * (3n + 1) is a whole number bigger than 1,
so I don't see what the problem is here.
 HW Helper P: 3,515 Haha yeah I thought about it while out today and realized the criteria n>2 is necessary, which goldust even incorporated into his proof! Sorry about that goldust.
 P: 85 Many thanks for the help! The proof is a bit trickier than I initially thought. When n is even and more than 2, n / 2 is an integer more than 1, and 3n + 1 is also an integer more than 1, so n / 2 * (3n + 1) ends up being divisible by both n / 2 and 3n + 1. Cheers!

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