TITLE:
How to Check If a Number Is Prime Using a Finite Definite Integral
AUTHORS:
Jesús Sánchez
KEYWORDS:
Primality Test, Number Theory, Primes, Factorization, Fourier Transform, Parseval’s Theorem, Time Domain, Frequency Domain, Numerical Computation
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.7 No.2,
February
22,
2019
ABSTRACT: In the history of mathematics
different methods have been used to detect if a number is prime or not. In this
paper a new one will be shown. It will be demonstrated that if the following
equation is zero for a certain number p,
this number p would be prime. And
being m an integer number higher than (the lowest, the most efficient the operation). . If the result is an integer, this result will tell
us how many permutations of two divisors, the input number has. As you can
check, no recurrent division by odd or prime numbers is done, to check if the
number is prime or has divisors. To get to this point, we will do the
following. First, we will create a domain with all the composite numbers. This
is easy, as you can just multiply one by one all the integers (greater or equal
than 2) in that domain. So, you will get all the composite numbers (not getting
any prime) in that domain. Then, we will use the Fourier transform to change
from this original domain (called discrete time domain in this regards) to the
frequency domain. There, we can check, using Parseval’s theorem, if a certain
number is there or not. The use of Parseval’s theorem leads to the above
integral. If the number p that we
want to check is not in the domain, the result of the integral is zero and the
number is a prime. If instead, the result is an integer, this integer will tell
us how many permutations of two divisors the number p has. And, in consequence information how many factors, the number p has. So, for any number p lower than 2m- 1, you can check if it is prime or not, just making the
numerical definite integration. We will apply this integral in a computer
program to check the efficiency of the operation. We will check, if no further
developments are done, the numerical integration is inefficient computing-wise
compared with brute-force checking. To be added, is the question regarding the
level of accuracy needed (number of decimals and number of steps in the
numerical integration) to have a reliable result for large numbers. This will
be commented on the paper, but a separate study will be needed to have detailed
conclusions. Of course, the best would be that in the future, an analytical
result (or at least an approximation) for the summation or for the integration
is achieved.