Is this a Pythagorean Theorem question?

Consider a sequence of increasing positive integers $\{ t_n \}_{n=1}^{n=\infty}$ such that $t_1^2 + t_2^2 + \cdots + t_m^2$ is a perfect square for all positive integers $m$ .

If $t_1 = 3$ , then the values of $t_2, t_3, t_4$ are all unique. However, $t_5$ is not unique because its possible values are 132, 204, 720 and 3612.

Prove that it is impossible to find a suitable value of $t_1$ such that all the subsequent terms ( $t_2, t_3, \ldots$ ) are unique.

P.S: I don't know whether I'm right or not. This is just a hunch.

#NumberTheory

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Comments

So it's pretty clear that $t_{m+1}$ is unique if and only if $t_1^2+\cdots+t_m^2$ is the square of a prime. Call that prime $p_m$ ; then $p_m = \frac{p_{m-1}^2+1}2.$ So the question is whether the recursive sequence $p_1 = p, p_m = \frac{p_{m-1}^2+1}2$ can ever produce only primes. E.g. for $p=3,$ the sequence goes $3,5,13,85,$ and then $p_4$ is not prime so $t_5$ is not unique.

This seems unlikely, but I don't have a proof. I did some numerical experiments; the only prime $p < 10^6$ for which the first five values of the sequence are all primes is $p=169219.$

Patrick Corn - 3 years, 7 months ago

What do you get for $t_2,t_3,t_4,t_5$ ? Am I making a mistake to say $t_2=4,t_3=12,t_4=84,$ and then $t_5$ is ambiguous? (Could be $204$ or $720$ or $3612$ ?)

I suspect you're right, and it shouldn't be too hard, but let me just make sure I'm not missing something first!

Patrick Corn - 3 years, 7 months ago

You're right. I've made a blunder. I've rectified the note above. Thanks.

Pi Han Goh - 3 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$