Some Assertions About Primes

A computer can quickly ``convince'' you that many assertions about prime numbers are true. Here are three.

• The polynomial $x^2+1$ takes on infinitely many prime values.
  Let
  $$f(n) = \{x : x < n : x$$    and $$x^2+1$$ is prime $$\}.$$
  With a computer, we quickly find that
  $$f(10^2) = 19, \quad f(10^3)=112, \quad f(10^4)=841, \quad f(10^5) = 6656.$$
  Surely $f(n)$ is unbounded! The PARI code to compute $f(n)$ is very simple:

  ? f(n) = s=0; for(x=1,n,if(isprime(x^2+1),s++)); s
  ? f(100)
  %1 = 19
  ? f(1000)
  %2 = 112
  ? f(10000)
  %3 = 841
  ? f(100000)
  %4 = 6656
  

• Every even integer $n>2$ is a sum of two primes.
  With a computer we find that this seems true

  $n$	$p$	$q$
  4	2	2
  6	3	3
  8	3	5
  10	3	7
  12	5	7

  ... and much further. In practice, it's easy to write an even number as a sum of two primes. Why should there be any weird even numbers out there for which this can't be done? PARI code to find $p$ and $q$:

  ? gb(n) = forprime(p=2,n,if(isprime(n-p),return([p,n-p])));
  ? gb(4)
  %7 = [2, 2]
  ? gb(6)
  %8 = [3, 3]
  ? gb(100)
  %9 = [3, 97]
  ? gb(1000)
  %10 = [3, 997]
  ? gb(570)          \\ takes no time at all!
  %11 = [7, 563]
  

• There are infinitely many primes $p$ such that $p+2$ is also prime.
  Let $t(n)=\char93 \{p : p \le n$ and $p+2$    is prime $\}$. Using a computer we quickly find that
  $$t(10^2)=8, \quad t(10^3)=35,\quad t(10^4)=205, \quad t(10^5)=1024.$$
  The PARI code to compute $t(n)$ is very simple:

  ? t(n) = s=0; forprime(p=2,n,if(isprime(p+2),s++)); s
  ? t(10^2)
  %12 = 8
  ? t(10^3)
  %13 = 35
  ? t(10^4)
  %14 = 205
  ? t(10^5)
  %15 = 1224
  

  Surely $t(n)$ keeps getting bigger!!

As it turns out, these three assertions are all OLD famous extremely difficult unsolved problems! Anyone who proves one of them will be very famous.

Assertion 2 is called ``The Goldbach Conjecture''; Goldbach reformulated it in a letter to Euler in 1742. It's featured in the following recent novel:

The publisher of that novel offers a MILLION dollar prize for the solution to the Goldbach conjecture:

  http://www.faber.co.uk/faber/million_dollar.asp?PGE=&ORD=faber&TAG=&CID=

The Goldbach conjecture is true for all $n<4\cdot 10^{14}$, see

  http://www.informatik.uni-giessen.de/staff/richstein/ca/Goldbach.html

Assertion 3 is the ``Twin Primes Conjecture''. According to

  http://perso.wanadoo.fr/yves.gallot/primes/chrrcds.html#twin

on May 17, 2001, David Underbakke and Phil Carmody discovered a 32220 digits twin primes record with a set of different programs: $318032361\cdot 2^{107001}\pm 1$. This is the current ``world record''.

With a computer, even if you can't solve one of these ``Grand Challenge'' problems, at least you can perhaps work very hard and prove it for more cases than anybody before you, especially since computers keep getting more powerful. This can be very fun, especially as you search for a more efficient algorithm to extend the computations.