When is 1+2+ +n a perfect square?

  • Thread starter Thread starter CornMuffin
  • Start date Start date
  • Tags Tags
    Square
CornMuffin
Messages
51
Reaction score
5

Homework Statement


Find the values of n\geq 1 for which 1! + 2! + ... + n! is a perfect square in the integers.


Homework Equations





The Attempt at a Solution


n=1 and n=3 works, but I don't know how to find anymore, or prove that there aren't anymore
 
Physics news on Phys.org
Do you know about quadratic residues? If so, that's a hint.
 
Petek said:
Do you know about quadratic residues? If so, that's a hint.

well I proved it, thanks
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
Back
Top