A proximate-prime polynomial is simply a quadratic equation - a finite polynomial of the second degree - that is derived from four successive (proximate, or neighboring) primes. Proximate-prime polynomials are interesting because they exhibit much greater prime densities than other polynomials.
When you graph primes against an X-axis that treats the expanding interval between successive perfect squares as a constant unit subdivided into equal parts, you produce a distinctive wave form for primes and prime factors.
For every composite number that is not itself a perfect square there exists a pair of nonconsecutive perfect squares whose difference is equal to the composite. Even before we get to the subject of factorization, the consequences of this observation are fascinating and far-reaching.
It began with an exploration of biquadratic paired primes: 2 primes separated by the equivalent of exactly 2 quadratic intervals.... Then the investigation took the logical next level by asking the question: Are there prime pairs that are separated by other, greater multiples of the quadratic interval? And if there are, what are the frequency characteristics by interval size and perfect square offset? The results are in, with charts, an Excel visualization, and masses of half-digested data...!
Find examples throughout this site that demonstrate using VBA code with worksheets and graphing - including generating primes, perfect squares, and composites, doing modular arithmetic, calculating GCDs, and more....
May 17th, 2013 Modal Distribution of Distinct Prime Factors: It’s not Necessarily What You Might Think
Yes, you can find the answers to life's most persistent questions here! What is the cumulative distribution of distinct prime factors? The answer is both what you were expecting and more than you were expecting. As expected, the cumulative frequency of unique prime factors diminishes with size, so that 2 is the most frequent factor, 3 is the second-most frequent factor, 5 the third, and so on for all prime factors. However, it turns out that the cumulative frequency of the largest prime factors - the biggest factor of each composite - has a modal distribution that is increasing at a slow rate approximating the cube root of N. Up to 10,000, this mode is 19. Up to 1,000,000, this mode is 73. It's fairly safe to assume that if we graph any given set of numbers, there will be a distinctive peak near P^3 = N.
Distribution of the largest prime factor for each composite less than 10,000. (The difference
the even and odd curves is accounted for entirely by the exclusion of prime numbers.)
July 7th, 2012 Paired, Squared, Cubed, and Primed...?
If we pick a square or a cube and count up to the nearest prime number, what are the odds that a prime number will also exist if we count up by the same amount from its root? The answer, it turns out, is surprisingly high - and even higher for cubes than for squares. If the question sounds confusing, it's easy to illustrate:
The results are:
For squared numbers up to 1 million, this relationship holds true 41.0% of the time (410 out of 1,000).
For cubed numbers up to 1 billion, this relationship holds true 59.8% of the time (598 out of 1,000).
Factoring in Polynomial Time: A Pronic Solution...
Sometimes the best things in life are free - well, almost free... and very simple... and blindingly obvious. A single GCD calculation using the closest pronic number to N will produce a factor for one-third of composites not divisible by 2 or 5 up to any size. For example, the nearest pronic to 898097881 is 898110992, and these numbers share a GCD of 1873 - a prime factor of both numbers.
An analysis of N < 10,000,000 shows that 35.8% of the nonobvious composites are factorable with a single GCD calculation. What is the common characteristic of this huge class of numbers? They appear to conform to rational angles in the Sacks number spiral. Such numbers can be generated with RadiusTest using the Lines option and produce polynomials with a third coefficient of 0.
Expanding the GCD calculation to pronic numbers within 6 quadratic intervals of N provides a nearly instantaneous factorization test for more than two-thirds of composites ending in 1, 3, 7, or 9 regardless of magnitude. Here is an analysis of composites less than 107 with GCDs calculated for pronics from 6 quadratic intervals less than N through 6 quadratic intervals greater than N. It shows a 74.4% success rate.
Desktop program and complete project source code for implementing the gold standard in primality testing. A fast and reliable test for numbers up to 1027-1 (that's 1 with 26 9s - a prime number...!).
(The project illustrates how to use a legacy language, VB6, not designed for big integers. It includes modular exponentiation code by DI Management Cryptography Software,)
"Fermatic" is a made-up word: Fermat + Automatic. This tool takes Fermat's great theorem to the limit, with some experiments to weed out pesky pseudoprimes. Rapidly generate prime, pseudoprime, and composite data.
Enter 3 or 4 numbers in a sequence and find out what the next 10, next 1,000, or next 10,000 values are. QTest lets you derive a quadratic equation from the values you input (and solve the equation's roots). Then you can use this polynomial to generate and analyze long number sequences for primality. (See Robert Sacks' method for quadratic derivation, used in Vortex.)
Calculate the products of infinite series using almost any inputs you can think of. Generates real number zeta series, demonstrating the calculation of many important constants - including e, pi, and phi, the Basel equality and Apéry's constant.
Tools that are digitally signed are published by: Wordwise Solutions
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theundesigned intelligence of the numberverse
Undesigned intelligence proposes that the universe is the way it is neither by accident nor by design. The axioms of mathematics, the laws of physics are what they are innately and without reference to percepted existent reality. The intelligence inheres in and is nonreferentially existent without regard to the observer. Undesigned intelligence is uncreated.