Let $k\ge 2$ be an integer. Can a product of $k$ consecutive integers be a perfect square?
Asked
Active
Viewed 342 times
0
-
@THELONEWOLF.: any as long as finite. – DeepSea Dec 11 '16 at 19:59
-
Let $k>0$ be an integer, can product of $k$ consecutive integer be a perfect square? – Dec 11 '16 at 19:59
-
5@THELONEWOLF.: You have vandalised this question! Changing "can" to "Can" is perhaps laudable, but surely unnecessary. Changing "doesn't" to "does not" is officious and annoying. Changing "perfect" to "peqrfect" is beyond comprehension. – TonyK Dec 11 '16 at 20:04
-
@123: You should edit that change into your question, rather than leaving it in a comment. – TonyK Dec 11 '16 at 20:07
-
1@123 why not prime numbers?? – Vidyanshu Mishra Dec 11 '16 at 20:09
-
Trivially, it works for $k = 1$ with the one integer being a square number. – Dan Dec 11 '16 at 20:11
-
1Any good theorem should have a proof without using primes, we have to learn to live without primes. —123 – Dec 11 '16 at 20:12
-
I have got something with prime, but since you do not want a proof with prime numbers then it is okay. – Vidyanshu Mishra Dec 11 '16 at 20:15
-
@6005 but i want a proof that does not using primes. – Dec 11 '16 at 20:15
-
By the way, This is good question. +1 – Vidyanshu Mishra Dec 11 '16 at 20:20
-
Possibly inspirational: http://math.stackexchange.com/questions/638646 – Bart Michels Dec 11 '16 at 20:21
-
2See also product of six consecutive integers being a perfect square for references and an idea of the difficulty of this question. – Bart Michels Dec 11 '16 at 20:23
-
Something tells me you won't be successful if it required heavy use of prime numbers to proof this by one of the brightest mathematician of 20th century, but well what do I know... – Sil Dec 11 '16 at 20:24
-
1@Sil, I am still thinking what problem OP has got with prime numbers, they are beautiful. – Vidyanshu Mishra Dec 11 '16 at 20:26
-
2@THELONEWOLF. Yea, using primes is nothing bad, I would say quite opposite. Primes are building blocks in number theory, seems quite unnatural to avoid them. I would understand avoiding using advanced theorems, but primes... – Sil Dec 11 '16 at 20:31
1 Answers
3
What you want is HERE $1$, and HERE $2$. I think copying/describing this text will take an entire hour which is not good for my fingers. So, just look at the paper I have given.
Note that the theorem in the paper $1$ is a generalised one. It states that The product of two or more consecutive positive integers is never a power. And your squares also come under this section.
Hope it will help you.
Vidyanshu Mishra
- 10,391
-
Firstly, thank you for your helpful comments. I have already read the article that you've linked, it is using primes. I want a proof that doesn't using primes, but thank you for your answer. – Dec 11 '16 at 20:53
-
-
2@123 Why don't you accept primes in the proof? This is almost as saying "I want a proof that doesn't use integers, but thank you for your efforts". – Dietrich Burde Dec 11 '16 at 22:25
-
I would agree with you @Dietrich Burde in this case. I m trying to convince OP in this matter for hours but no result. – Vidyanshu Mishra Dec 11 '16 at 22:55
-
The article that you've linked after the first one is helped me. This answer is what I'm looking for! – Dec 12 '16 at 13:22
-