1

Suppose $a$ is a positive integer $(2a+1)$ and $(3a+1)$ are perfect squares. Prove that $(5a+3)$ is a composite number. I really do not know the relationship between perfect squares and composite numbers.

Achari S Ganesha
  • 898
  • not complex number, composite, @barakmanos – Adam Hughes Jul 12 '14 at 09:43
  • Is the following related : http://math.stackexchange.com/questions/575733/if-4n1-and-3n1-are-both-perfect-sqares-then-56n-how-can-i-prove-this – lab bhattacharjee Jul 12 '14 at 09:44
  • Formula there. Using the Pell equation can write the solution of this equation and then the expression itself. http://math.stackexchange.com/questions/756420/diophantine-equation-x2-py2-fracp-12/825618#825618 – individ Jul 12 '14 at 09:53

1 Answers1

4

$2a+1=m^2$ and $3a+1=n^2$ for some positive integers $m,n$. Then $3m^2-2n^2=1$. Also $5a+3=(2a+1)+(3a+1)+1=m^2+n^2+1=m^2+n^2+(3m^2-2n^2)=4m^2-n^2$. This is a difference of two squares, hence a product. I leave it to you to show that neither factor is 1, and hence $5a+3$ is composite.

paw88789
  • 40,402