neudateaLp

2022-11-22

Can a dot product of a permutation of n (1,-1) with a sequence of primes generate unique numbers?
According to Fundamental Theorem of Arithmetic any positive whole number is the product of primes. Therefore, I can create unique numbers by multiplying n primes.
If I have a list of length n generated by a finite sequence in the set 1,-1, e.g., (1,1,-1,-1,1,...), and I do a dot product with a sequence of n primes starting with 3, e.g., (3,5,7,11,13,...), do I have any guarantee that I will generate unique numbers by doing different permutations of 1s and -1s?
I know that I can't if my prime's sequence start with 2, $\left[1,1,-1\right]\cdot \left[2,3,5\right]=\left[-1,-1,1\right]\cdot \left[2,3,5\right]$, but I'm not sure for the cases where my sequence starts with higher primes.

Monserrat Wong

Expert

Step 1
No, there is no such guarantee. Some quick Python code shows that, for [3,5,7,11,13] we have:
$7=3\left(1\right)+5\left(-1\right)+7\left(1\right)+11\left(-1\right)+13\left(1\right)$
and
$7=3\left(-1\right)+5\left(1\right)+7\left(1\right)+11\left(1\right)+13\left(-1\right)$
Step 2
This is enough to disprove your claim, but this is just one counter-example, I found many others with my code.

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