how to prove the infinitude of twin primes?
Answers
The infinitude of twin primes can be proved by the simple steps.
Let the number of twin primes be finite in number and P and Q be the largest pair of twin primes.
Now we know that there are infinite number of prime numbers which also includes P and Q.
Take the product of all the primes.
Let it be x.
Now, we add 1 to x we get a prime number and if we substract 1 from x we get a prime number.
Therefore we have got two prime numbers with difference of 2. By definition of twin primes we have Any two primes with difference 2 are twin primes.
Therefore we have got a pair of twin primes which is greater than P and Q. Therefore our assumption that twin primes are finite is incorrect.
This proves that twin primes are infinite.
Well this theorem does not stand true as my thinking because I believe that the number line is curved which implies that the numbers are finite. The purpose for my believing so is the geometry of the universe which is elliptical (spherical) geometry which then implies that nothing in this universe is straight. Every straight line is a part of circle with infinite circumference. And hence the number line is also curved which straightaway disproves the infinitude of twin primes, polignac conjecture and other such theorems.