BEAUTY OF CONSICUTIVE NUMBERS

[MATHPRITOM]

Find all the natural number x such that $x(x+1)(x+2)(x+3)(x+4)=k^2.$

[sm.joty]

No solution may be. except 0

[Phlembac Adib Hasan]

No solution may be. except 0

Yes, it has no solution.It follows from a theorem of Paul Erdos.

[Masum]

Hmm. Actually it is very dangerous to use these kind of theorems(For more example: Dirichlet's Theorem, Fermat's Theorem, Catalan's conjecture(proved), Bertrands postulate, Erdos'-Seilberg's theorem and so more). Because their proofs are not that easy or not all are elementary. Specially unless you understand the proof of the theorem. Some theorems have recently been used widely without proofs, such as LTE or Zsigmondy's Theorem. Though Zsigmondy's theorem's proof uses Cyclotomic polynomial's idea, it has become very popular. However, try to avoid them as far as you can, but search in google for their proofs.
In this particular type of problems like proving that a natural number is not a perfect power, the following ideas come to the rescue generally:
1. Write it as $ab$ where $a,b>1,\gcd(a,b)=1$. So that you have $a=m^k,b=n^k$ and then seek for some contradiction.
2. The maximum power of a prime $p$ in that number must be divisible by $k$, the exponent(LTE and Zsigmondy may help you in this regard).
3. Or you may try to show that it lies between two consecutive $k-th$ powers. Even you may have that between $(a-2)^k$ and $a^k$ and you can conclude that is equal to $(a-1)^k$.
4. Use congruences of different numbers yielding more information.\
First try proving that, $x(x+1),x(x+1)(x+2)$ and $x(x+1)(x+2)(x+3)$ are not perfect powers.

[Masum]

Another problem regarding these tactics.
viewtopic.php?f=15&t=788