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

Question

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&#039;t if my prime&#039;s sequence start with 2,   [  1  ,  1  ,  −  1  ]  ⋅  [  2  ,  3  ,  5  ]  =  [  −  1  ,  −  1  ,  1  ]  ⋅  [  2  ,  3  ,  5  ], but I&#039;m not sure for the cases where my sequence starts with higher primes.

Monserrat Wong · Accepted Answer

Step 1No, 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 2This is enough to disprove your claim, but this is just one counter-example, I found many others with my code.