Question #19870

Prove that a prime number cannot be expressed as the sum of two or more consecutive
positive odd integers.

Expert's answer

if we have p prime and corresponds p=a1+a2+...+an leta1=2a+1 then a2=2a+3 and so on ai=2a+2i-1 we know that sum( i=1 to n)

(i)=(1+n)*n/2 we have p=sum( i=1 to n) (2a+2i-1)=(2a-1)*n+sum( i=1 to n)

(2i)=(2a-1)*n+2*sum( i=1 to n) (i)=(2a-1)*n+2*(1+n)*n/2=

=(2a-1)*n+(1+n)*n=n*(n+2a-1)

and we see that n must divide p and it means that forn>1 we have that p not prime but it is contradiction so p cant be expressed

as the sum of two or more consecutive positive odd numbers

(i)=(1+n)*n/2 we have p=sum( i=1 to n) (2a+2i-1)=(2a-1)*n+sum( i=1 to n)

(2i)=(2a-1)*n+2*sum( i=1 to n) (i)=(2a-1)*n+2*(1+n)*n/2=

=(2a-1)*n+(1+n)*n=n*(n+2a-1)

and we see that n must divide p and it means that forn>1 we have that p not prime but it is contradiction so p cant be expressed

as the sum of two or more consecutive positive odd numbers

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