## A Study of the Sum Three or More Consecutive Natural Numbers

**Authors:** Emmanuil Manousos

It holds that every product of natural numbers can also be written as a sum. The inverse does not hold when 1 is excluded from the product. For this reason, the investigation of natural numbers should be done through their sum and not through their product. Such an investigation is presented in the present article. We prove that primes play the same role for odd numbers as the powers of 2 for even numbers, and vice versa. The following theorem is proven: “Every natural number, except for 0 and 1, can be uniquely written as a linear combination of consecutive powers of 2 with the coefficients of the linear combination being -1 or +1.” This theorem reveals a set of symmetries in the internal order of natural numbers which cannot be derived when studying natural numbers on the basis of the product. From such a symmetry a method for identifying large prime numbers is derived. We prove a factorization test for the natural numbers.

**Comments:** 33 Pages.

**Download:** **PDF**

### Submission history

[v1] 2018-09-23 10:45:03

[v2] 2018-10-15 13:52:47

[v3] 2018-12-12 15:48:04

**Unique-IP document downloads:** 80 times

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