# Prime numbers vs consecutive natural numbers.

#### mente oscura

##### Well-known member
An easy question.

All "odd" number can be expressed as a sum of consecutive natural numbers.

Example:

$$35=17+18$$

$$35=5+6+7+8+9$$

$$35=2+3+4+5+6+7+8$$

Question:

Demonstrate that prime numbers (except for the "2"), can only be expressed as the sum of two consecutive natural numbers.

#### mathbalarka

##### Well-known member
MHB Math Helper
Re: prime numbers vs consecutive natural numbers.

Elementary. Sum of $k$ consecutive natural numbers is either $0 \pmod{k}$ or $0 \pmod{k/2}$ so the only plausible candidates are $k = 1$ and $k = 2$ which is easy to verify for odd primes.

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#### mente oscura

##### Well-known member
Re: prime numbers vs consecutive natural numbers.

Sum of $k$ consecutive natural numbers is $0 \pmod{k}$ so the only plausible candidate is $k = 1$ which is easy to verify for odd primes.
$$7=3+4 \rightarrow{} k=2$$

#### mathbalarka

##### Well-known member
MHB Math Helper
Re: prime numbers vs consecutive natural numbers.

Look at it again.

##### Well-known member
the question should be
Demonstrate that only prime numbers (except for the "2"), can be expressed as the sum of two consecutive natural numbers only.

let the number of numbers be n and 1st number a+1

then sum of numbers= an + n(n+1)/2

it is integer
if n is odd (n+1)/2 is integer so it is divsible by n

if n is even an and n(n+1)/2 is divisible by n/2

so if n > 2 and odd it is not prime as divsible by n

if n > 2 and even it is divisible by n/2(which is >= 2) so not prime

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