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Putting this in your first condition $pp_n+p=p_n\Rightarrow p(p_n+1)=p_n$$\Rightarrow p=\frac{p_n}{p_n+1}\Rightarrow 0<p<1$ therefore no such prime is possible

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This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

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## Comments

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TopNewest@Yajat Shamji, I did my best. The simplest proof

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Ok. What about @Mahdi Raza?

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I think that @Mahdi Raza won - check Sum of squares.

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Did you ask that question to be created as a note @Yajat Shamji?

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@Yajat Shamji

Fun factI and Mahdi Raza answered at the same time

Check the time on both of our

comments/notesLog in to reply

$1$ day ago...

Yes, I checked - it wasYou both win?...

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How about giving us another challenge @Yajat Shamji...?

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What about this?:

Question: If $x + n + p$ and $p(x + n)$ both equal a prime $p_n$, prove that $\frac{x + n}{p}$ is also the same prime $p_n$.

$x + n$ is an integer and $p$ is a prime.

This time, though, your challenge is with @Zakir Husain

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Case 1 $\alpha$ is even:If $p>2\Rightarrow p$ is odd

$\Rightarrow p\alpha$ is even therefore it can never be prime

If $p=2$

$\Rightarrow p\alpha$ is divisible by $4$

Case 2 $\alpha$ is odd:If $p$ is odd

$p+\alpha$ is even therefore it can never be prime

If $p=2$

$\Rightarrow p\alpha$ is $2,x+n=1$ or otherwise composite

$\therefore$ $x+n=1$ is the only possible value

Now for $x+n=1;p_n=\frac{1}{p}\Rightarrow p_n \cancel{\in} Z$

therefore no answer is possible.

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$\frac {x+n}{p}=p_n\Rightarrow x+n=pp_n$

Another proof letPutting this in your first condition $pp_n+p=p_n\Rightarrow p(p_n+1)=p_n$ $\Rightarrow p=\frac{p_n}{p_n+1}\Rightarrow 0<p<1$ therefore no such prime is possible

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$1$ of the challenge (albeit the only one) - make a note on it - I'd advise using both proofs...

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@Zakir Husain beat you to it...

I thinkLog in to reply

@Yajat Shamji!

Yup, but I am sorry that I didn't read this comment till upto nowLog in to reply

It seems as if I have a challenge for you...!

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$1$ to $2$ hours before...

I got it incorrect anyways. I attempted itLog in to reply

@Yajat Shamji!

Oh... Sorry to hear thatLog in to reply

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