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a) Can we establish a proof, there exists infinitely many primes of the form $n^2$ + 1. Why is the unit digit of such a prime p always 1 or 7? Is there any reasonable procedure or concept for the fact that the unit digit 7 occur essentially twice as often, when we identify the primes < 10000?

b) can we prove or disprove that, there exists an interval of the form [$n^2$, $(n+1)^2$] containing at least 1000 prime numbers.

c) We know that even integer > or = 4 can be written as sum of two primes and integers > 5 can be written as sum of three primes. Of course, those are conjectures. I am not asking the proof of those conjectures. I would like to know those statements are equivalent or not. If yes, how you will justify?

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    $\begingroup$ It has not been established that there are infinitely many primes of the form $n^2+1$. What does "Inetvel" mean? $\endgroup$ – Gerry Myerson Nov 8 '12 at 11:40
  • $\begingroup$ what's "inetvel" ? $\endgroup$ – Fattie Jul 10 '16 at 12:24
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If every even $n\ge4$ can be written as a sum of two primes, then every integer $m\ge6$ can be written as either $2+r$ or $3+r$ where $r\ge4$ is even, hence $m$ can be written as a sum of three primes.

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A partial answer:

a) The unit digit is always 1 or seven because it is not possible to end up with a number of the form $n^2+1$ with a unit digit of 9 or 3, and 5, while possible, implies that the number is dividable by 5.

b) Plug in $n=1000000$ (It works as the interval).

c) They have nothing (intrinsically) to do with each other.

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Answer to $(a)$: If the unit digit of $n$ is an odd number , then $n^2+1$ is divisible by $2.$ If the unit digit is $2$ or $8$ , then $n^2+1$ is divisible by $5.$ If the unit digit is $0$ ,then the last digit of $n^2+1$ is $1.$ If the unit digit is $4$ or $6$ , then the last digit of $n^2+1$ is $7.$

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  • $\begingroup$ Please find out how to type math on this site. The first principle is to surround mathematical notation with dollar signs, $ $. No need to sign your name, either as it shows up beneath the post. $\endgroup$ – Kevin Carlson Nov 8 '12 at 11:36

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