MHB Is the Converse of the Given Statement True for Any Positive Integer n?

[johnny009]

if n is a positive integer greater than 2 and m the smallest integer greater than or = n, that is a perfect square.
Let a = m-n.

Show that if n is prime, then a is not a perfect square.

Also, is the converse of above true, for any integer n?

any guidance, will be much appreciated?

Thanks

 

[I like Serena]

if n is a positive integer greater than 2 and m the smallest integer greater than or = n, that is a perfect square.
Let a = m-n.

Show that if n is prime, then a is not a perfect square.

Also, is the converse of above true, for any integer n?
any guidance, will be much appreciated?Thanks

Hey johnny009! Welcome to MHB! (Smile)Guidance: let's try a couple of examples, starting with the simplest we can think of.The smallest prime $n$ is $3$, in which case $m=2^2=4$, and $a=4-3=1$, which is a perfect square!
Ah well, maybe $a=1$ is a special case...

Let's try again, the next prime $n$ is $5$, so that $m=3^2=9$, and $a=9-5=4$, which is again a perfect square!

Erm... I think it's not true, and we have 2 counter examples to prove it.Continuing with $n=6$, we get $m=3^2=9$, and $a=9-6=3$, which is not a perfect square... and $n$ is not prime.
So we have a counter example for the converse as well.

 

[johnny009]

Hey johnny009! Welcome to MHB! (Smile)Guidance: let's try a couple of examples, starting with the simplest we can think of.The smallest prime $n$ is $3$, in which case $m=2^2=4$, and $a=4-3=1$, which is a perfect square!
Ah well, maybe $a=1$ is a special case...

Let's try again, the next prime $n$ is $5$, so that $m=3^2=9$, and $a=9-5=4$, which is again a perfect square!

Erm... I think it's not true, and we have 2 counter examples to prove it.Continuing with $n=6$, we get $m=3^2=9$, and $a=9-6=3$, which is not a perfect square... and $n$ is not prime.
So we have a counter example for the converse as well.

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Hi There,

Thanks a lot for the reply.

But, your solutions ignores the fact, that 'm' cannot be less than 'N' ...as per the QUESTION??

So, your solution...is not really addressing the Question.

CHEERS

John.

 

[I like Serena]

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Hi There,

Thanks a lot for the reply.

But, your solutions ignores the fact, that 'm' cannot be less than 'N' ........as per the QUESTION??

So, your solution...is not really addressing the Question.

CHEERS

John.

I'm assuming you mean 'n' instead of 'N', since there is no reference to 'N'?
Erm... in each of the examples $m\ge n$ as per the question... am I missing something? (Wondering)