Perfectly square factorials

Number Theory Level 2

How many elements of the set $A=\{1!,2!,3!,\ldots,2015!\}$ are perfect squares?

44 6 15 1

3 solutions

Janardhanan Sivaramakrishnan
Jul 11, 2015

$1!$ is a perfect square. and $2!$ is not. For every $n>2$ , by Bertrand's postulate there is a prime number $p$ such that $n/2 < p < n$ . Therefore, the prime factorization of $n!$ would have the radix of $p$ as 1.

This would mean that $\sqrt{n!}$ would be irrational. Thus, there is only one value of $n$ for which $n!$ is an integer.

Mistake in the last line it should be square root of n!

Kushagra Sahni - 5 years, 10 months ago

Department 8
Jul 12, 2015

Such an easy question the number $n!$ will be a perfect square if $n$ repeat twice at minimum [because any $n!$ can be perfect of each number repeat twice]. So there is only one number that satisfy this. $1!$

A number is a perfect square if each prime dividing it repeats an even number of times. Consider $36=9\cdot 4$ . Does $9$ come two times? No it doesn't. But each prime dividing $36$ e.g. $2$ and $3$ appear twice in the prime factorization.

Rahul Saha - 5 years, 11 months ago

Rajdeep Ghosh
Jul 21, 2015

1 ka factorial sirf odd ... Baki sab 2,6,24,120 ...so on r evn

So, what will happen if they are even?

Swapnil Maiti - 5 years, 10 months ago

Perfectly square factorials

3 solutions

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