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Let's say that I want to calculate the sum of 20 prime numbers that reside in the range up to 1000. Will the sum (addition, not multiplication) of any 20 primes in that range be unique , or will I see recurrences?

Robert Z
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azer
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1 Answers1

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Note that $3+13=5+11$, so you have no uniqueness.

P.S. Note that by looking at A045917, you can even find 40 distinct primes $p_1,\dots, p_{40}$ such that $$\sum_{k=1}^{20}p_k=\sum_{k=21}^{40}p_k.$$ For example the sum of $$3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 131, 137, 139, 149, 151, 157, 163, 167, 173, 239$$ is equal to the sum of $$37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 109, 113, 211, 227.$$

Robert Z
  • 145,942
  • Thank you. Multiplications will yield unique results in such case, correct? that is, if any of the 20 prime numbers is used only once – azer Sep 28 '16 at 07:40
  • @azer Yes because any integer has a unique factorization. – Robert Z Sep 28 '16 at 07:42