MHB Prime numbers vs consecutive natural numbers.

AI Thread Summary
All odd numbers can be expressed as the sum of consecutive natural numbers, while prime numbers (except for 2) can only be represented as the sum of two consecutive natural numbers. The sum of k consecutive natural numbers results in specific modular conditions, limiting viable candidates for k to 1 or 2. This leads to the conclusion that odd primes can be expressed as sums of two consecutive numbers, such as 7 = 3 + 4. The discussion emphasizes the unique property of prime numbers in relation to sums of consecutive natural numbers.
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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+8Question:

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

mathbalarka said:
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
 
Re: prime numbers vs consecutive natural numbers.

Look at it again.
 
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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