# Prove that two sequences of integers that have the same sum and product must be the same.

Given two sequences of nondecreasing distinct positive integers such that $$x_1 + x_2 + ... + x_i = y_1 + y_2 + ... + y_i , i>0$$ and that $$x_1x_2 ... x_i = y_1y_2 ... y_i$$ Prove/disprove that the sequences are equal i.e. $$x_1 = y_1, x_2 = y_2, ... , x_i = y_i$$

I started with
Let $$x_1x_2 ... x_i$$ be $$A$$. If $$A$$ is prime, $$x_1 = A = y_1$$ (since $$A$$ cannot be factored any more) and we are done.

What I don't know is what happens when $$A$$ is not prime. Intuitively, it sounds true, and I cannot find any counter examples.

• $x_1=1, x_i=0 \forall i \ne 1$ and $y_2=1, y_i=0 \forall i\ne 2$
– user563311
Oct 29, 2018 at 8:40
• @JoeyDoey the Xs and Ys are positive integers Oct 29, 2018 at 9:17
• You should state in your question that $x_i \gt 0$. Positive vs non-negative is a source of confusion. I'm pretty sure I've been told in school (France) that positive is $\ge 0$ while in English, people use "non-negative" to describe this. This is a case where symbols actually make thing clearer. Anyway, your statement is at least false because of permutation. $y_0 = x_i \land \forall i \ne 0, y_i = x_{i-1}$ Oct 29, 2018 at 9:36
• Donald, you should at least require that the sequences $(x_i),(y_i)$ are non-decreasing. That way, William Elliot's counterexample is ruled out. Oct 29, 2018 at 10:00
• @TonyK thank you for the tip, edited. Oct 29, 2018 at 16:59

Counterexample:

$$12+4+3 \ =\ 9+8+2$$

$$12\cdot4\cdot3 \ = \ 9\cdot8\cdot2$$

Moreover, for $$\ i>2\ ,\$$you can always find infinitely many counterexamples.

• Welcome to Math.SE! It would improve your answer to explain the existence of "infinitely many counterexamples" in more detail (and perhaps add a remark about solutions with $i=2$). Oct 30, 2018 at 15:25
• This answer saves me the trouble of trying to find a proof for $i > 2$. I removed my answer because the missing full proof I'm trying to find doesn't exist. +1. Oct 30, 2018 at 23:48
• Very good really. Oct 31, 2018 at 0:53
• @hardmath with $12, 4, 3$ and $9, 8, 2$ a counterexample, $12a, 4a, 3a$ and $9a, 8a, 2a$ with $a > 0$ is a counterexample too, since $a(12 + 4 + 3) = a(9 + 8 + 2)$ and $a^3 \cdot 12 \cdot 4 \cdot 3 = a^3 \cdot 9 \cdot 8 \cdot 2$. That makes an infinite number of counterexamples in case $i = 3$ Oct 31, 2018 at 7:34

Let $$x_1 = y_2, x_2 = y_1$$ for a quick counter example.

• Since the sequences must be non-decreasing, this substitution only works if $x_1=x_2$ and $y_1=y_2$, in which case it is not a counter-example. Oct 29, 2018 at 17:17
• @M.Nestor: the question has changed. Oct 29, 2018 at 18:35