neudateaLp

Answered

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, $[1,1,-1]\cdot [2,3,5]=[-1,-1,1]\cdot [2,3,5]$, but I'm not sure for the cases where my sequence starts with higher primes.

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, $[1,1,-1]\cdot [2,3,5]=[-1,-1,1]\cdot [2,3,5]$, but I'm not sure for the cases where my sequence starts with higher primes.

Answer & Explanation

Monserrat Wong

Expert

2022-11-23Added 7 answers

Step 1

No, there is no such guarantee. Some quick Python code shows that, for [3,5,7,11,13] we have:

$7=3(1)+5(-1)+7(1)+11(-1)+13(1)$

and

$7=3(-1)+5(1)+7(1)+11(1)+13(-1)$

Step 2

This is enough to disprove your claim, but this is just one counter-example, I found many others with my code.

No, there is no such guarantee. Some quick Python code shows that, for [3,5,7,11,13] we have:

$7=3(1)+5(-1)+7(1)+11(-1)+13(1)$

and

$7=3(-1)+5(1)+7(1)+11(1)+13(-1)$

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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