exploring the undesigned
intelligence of the numberverse
lists/generators: Perfect Squares, Composites, Primes
The First 1,000,000...
Generate Your Own Up to 1027...
Secrets of the Spiral
Proximate-Prime Polynomials
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.
The high primality of prime-derived quadratic sequences
What is a "perfect prime polynomial"?
What is polynomial pronic alignment?
Use an Excel spreadsheet to explore polynomial primes!
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.
Try out an ingenious Excel worksheet to see the wave!
Counting primes by quadratic interval
How do you make Fermat's method of factorization faster than trial division...?
The Magic Square of Subtraction...
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.
Part 1 - A 'Classic Discovery'
Part 2 - A Deterministic Test
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...!
Read about the latest findings
What are Biquadratic Paired Primes?
Two "new" rules of perfect squares
How I learned not to be afraid of big numbers
It's a question of magnitude...
Using Excel as a tool for number theory
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....
November 17th, 2011
What has 6 got to do with it?
Here is a little discovery: The difference between the first and fourth prime number of a proximate-prime polynomial is ALWAYS A MULTIPLE OF 6.
For example:
Prox. Prime Poly. |
1st Term |
4th Term |
4t-1t |
7th Term |
7t-1t |
10th Term |
10t-1t |
13th Term |
13t-1t |
n^2 + n + 10157 |
10159 |
10177 |
18 |
10213 |
54 |
10267 |
108 |
10339 |
180 |
n^2 - n + 10331 |
10331 |
10343 |
12 |
10373 |
42 |
10421 |
90 |
10487 |
156 |
2n^2 - 2n + 10627 |
10627 |
10651 |
24 |
10711 |
84 |
10807 |
180 |
10939 |
312 |
n^2 - n + 11777 |
11777 |
11789 |
12 |
11819 |
42 |
11867 |
90 |
11933 |
156 |
n^2 - n + 12107 |
12107 |
12119 |
12 |
12149 |
42 |
12197 |
90 |
12263 |
156 |
2n^2 - 2n + 12277 |
12277 |
12301 |
24 |
12361 |
84 |
12457 |
180 |
12589 |
312 |
2n^2 - 2n + 12409 |
12409 |
12433 |
24 |
12493 |
84 |
12589 |
180 |
12721 |
312 |
3n^2 - 3n + 12653 |
12653 |
12689 |
36 |
12779 |
126 |
12923 |
270 |
13121 |
468 |
n^2 + 5n + 12785 |
12791 |
12821 |
30 |
12869 |
78 |
12935 |
144 |
13019 |
228 |
n^2 + n + 12887 |
12889 |
12907 |
18 |
12943 |
54 |
12997 |
108 |
13069 |
180 |
Surprisingly, this divisibility by 6 does not stop with the fourth term. It recurs with the polynomial's 7th term, 10th term, 13th term, and so on ad infinitum.
Lots more data available: PPPs < 50000, t1 - t15 ("T" denotes each t value divisible by 6) Download (115KB)
Do you know why?
This is probably related to this observation about perfect squares and, possibly, the Euler and Riemann zeta functions.
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. Analysis for N<107 Download (11MB) Previous Articles |
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Tools for Testing Natural Numbers |
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MinusSquare |
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Miller-Rabin |
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FermaticTest |
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QTest |
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RadiusTest |
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ZetaTest |
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PrimeTest |
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MinusSquare: A New Factoring Algorithm (beta) |
The magic square of subtraction has given birth to a baby factoring algorithm.
Download Program (25KB) |
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Miller-Rabin Demo with source code (beta) |
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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...!).
Download Project (10KB) Download Program (17KB) |
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FermaticTest (beta) |
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"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.
Download (34KB) |
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QTest (beta) |
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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.)
Download (35KB) How to Use QTest |
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RadiusTest (beta) |
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Calculate prime and composite distribution in the Sacks Number Spiral by offset (curved series) and angle (straight series).
Download (22KB) |
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ZetaTest (beta) |
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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.
Download (23KB) |
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PrimeTest (beta) |
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Prime and composite number calculator (with prime and nonprime factor analyzer and prime number generator).
Download (31KB) |
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Tools that are digitally signed are published by: Wordwise Solutions |
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