Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
1  
It has not been established that there are infinitely many primes of the form $n^2+1$. What does "Inetvel" mean? –  Gerry Myerson Nov 8 '12 at 11:40

3 Answers 3

up vote 1 down vote accepted

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.

share|improve this answer

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 $7.$

share|improve this answer
    
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. –  Kevin Carlson Nov 8 '12 at 11:36

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.