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Searching for Prime

Classical primality testing of large numbers requires a number to be rigorously divided by all prime numbers, up to the square root of the number to be tested. This method is timeand resourceconsuming for large numbers. Some time is gained by only dividing by prime numbers to determine the factors, but this too falls short where large numbers are tested for which not all the required lower primes are known. The problem becomes no easier when the very next higher prime number is sought, because the entire rigorous process has to be repeated for every number, and the number of calculations increase as the numbers get bigger. A prime number is a positive integer that can only be fully divided by 1 and itself. Regarding primality, a positive integer can only have 2 states: prime or non-prime. It is one or the other, no inbetweens. If it can be proven that a number is not non-prime, it is inherently proven that it is prime. This is also the basis upon which the Sieve of Eratosthenes works [1]. A method is presented for finding prime numbers, “from the bottom up” thereby allowing the sieve to be expanded as new prime numbers are discovered. The number of calculations required is greatly reduced by using a quadratic relationship. Introduction: For centuries man has been fascinated by prime numbers, forever trying to find patterns or series that may reveal the next unknown prime number. In 1772 Euler identified a string of 40 prime numbers described by the equation [2] y = 41 + n + n (for n = 1 ... 40) ( 1 ) I have tried to better this and so far have failed. The Sieve of Eratosthenes is an ancient method by which all possible non-primes are identified, by “quickly” testing for integer factors. Numbers left behind have no factors, and are therefore accepted to be prime. The test is enhanced by testing only for prime factors, since all composite factors are also products of prime numbers. To test large numbers, all primes below the square root of the highest checked number must be known, or else it needs to be checked for every odd number, from the last known prime, up to the said square root. This method becomes cumbersome when large sets of numbers are checked, due to the increased number of calculations required. A single number can be checked for primality by testing for integer factors. Dividing by every prime number lower than the square root of the tested number, with no integer result, proves the number to be prime. Once again this becomes more cumbersome if all the prime numbers below the square root of the tested number is not known. A multitude of other specialized methods have been developed to determine very large “special” primes. By these I mean numbers that answer to a specific set of rules, e.g. a-1 and other structures for which there are many excellent references available [3]. The most recent great successes have been by the GIMPS project, in the detection and proving of largest prime numbers as Mersenne Primes. [4]. Mersenne primes are of the form 2-1. Other special primes with unique probability tests have been found [5], however, even putting all these primes together still leaves huge gaps of unidentified primes. It becomes more and more difficult to identify and prove these unknown primes, since not all lower order primes are known. There is a saying in cycling terms that if you focus too much on the rock in the path, you will hit the rock and fall off your bicycle. I am a poor cyclist, before anyone asks, but I have decided to look where prime is not. I present a new method of finding prime numbers. I believe I have found a practical, provable method of identifying and eliminating all non-prime numbers, therefore by default proving remaining numbers to be prime. If it is not non-prime, it is prime! Theorem: There exist a series of equations ym = n 2 + m ∗ n for any n e (1, 2, 3...) and any m e (0, 1, 2, 3...) where all values of y is non-prime, except for values at n=1 for which the value of y might be prime. Any value of y appearing at any n>1, for any of the equations (ym), shows that value as being nonprime and having at least one factor of n. Any value of y appearing at n=1, for any of the equations (ym), that have not been identified as nonprime by any prior series (all lower values of m, expanded up to and past the value of y in question), thereby having no factors, must be prime. Proof: ym = n 2 + mn ( 2 )