The End of Integers

Euclid in 300 BC proved one of the beautiful theorems of mathematics that the number of prime numbers is infinite. The proof is based on a simple argument. Suppose $p_1$ , $p_2$ , $\ldots$ , $p_r$ be all the primes. Let $P$ be the product of all the primes, then $P+1$ is either a prime or a product of primes which are greater than those listed. In either case, a new prime is discovered. Here the implicit assumption is that integers are assumed to lie on a number line — the so-called real line.

I recently uploaded a paper on ResearchGate in which I showed that real line does not exist but numbers lie on a circle. You can find the paper here. I showed that there are only finitely many primes. I showed that the product of all primes is, in fact, the period of integers. If $P=p_1p_2\ldots p_r$ , then $P+n=n$ , where $n$ is a positive integer. It shows that $P$ is the largest integer after which the integers loop back to the beginning.

P.S. We note that if there exists an upper bound $P$ , then there also exists a lower bound $l_q = 1/P$ , which is the least quantum of numbers. From a physics point of view, one might wonder that this length is the Planck length $l_p = 1.6 \times 10^{-35}$ m. But this is misleading because the Planck’s length has the unit of length whereas $l_q$ is dimensionless. But knowing $l_q$ is very important. It would guide us what is the largest prime number.

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