Pull back the curtain on infinity

exploring the undesigned
intelligence of the numberverse

lists/generators: Perfect Squares, Composites, Primes

 

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The First 1,000,000...

Lists of Perfect Squares

Lists of Composites

Lists of Primes

Generate Your Own Up to 1027...

Perfect Square Generator

Prime Factors Generator

Prime Generator

Experimental Calculators

ZetaTest: Infinite series

MinusSquare: Factoring

FermaticTest: Probable prime

QTest: Quadratic Series

RadiusTest: Spiral lines and curves

PrimeTest: Analysis

Miller-Rabin: Demonstration

Secrets of the Spiral


Explore the Sacks Number 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!

 

Catch the Wave...

The Wave

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

 

Fixing Fermat’s Factorization Method

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
 

 

Q-Paired Primes...

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....

June 4th, 2013
Factor Families

A factor family is a set of composites that share common factors. The family begins with a semiprime, and all members of the set share the two prime factors of the semiprime. The family can be extended indefinitely. For example, for the first 100,000 odd composites the factor family of 4619 (149*31) looks like this:

There are a few things to note about this representation: The prime factors (in red) are distinct, so, for example, 3 can be 3*3 or 3*3*3, and so on. The complete factors (nonprime and prime in blue) of each composite all have three common factors: the semiprime and its prime factors. By definition, a semiprime cannot have nonprime factors.

Download Excel worksheet: Factor Families (1.5 MB)

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
between the even and odd curves is accounted for entirely by the exclusion of prime numbers.)

Download Excel worksheet: Distinct Factor Analysis (macro-enabled) (30KB)

 

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:

Squared example:

Square Prime Offset Root Prime Offset
3025 3037 +12 55 67 +12

Cubed example:

Cube Prime Offset Root Prime Offset
79507 79531 +24 43 67 +24

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).

Download CSV files: Squared-1000 (30KB) and Cubed-1000 (35KB)

  • For squared numbers up to 100 million, this relationship holds true 31.4% of the time (3,143 out of 10,000).
  • For cubed numbers up to 1 trillion, this relationship holds true 44.7% of the time (4,472 out of 10,000).

Download CSV files: Squared-10000 (351KB) and Cubed-10000 (425KB)

 

November 17th, 2011
What has 6 got to do with it?

Here is a little observation: 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?

 

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
Instantly factor a semiprime!
Reveal the DNA of semiprimes
The Q square of factoring

 

Tools for Testing Natural Numbers

 MinusSquare
 Miller-Rabin
 FermaticTest
 QTest
 RadiusTest
 ZetaTest
 PrimeTest
 
MinusSquare: Alternative Factoring Algorithm (beta)

The magic square of subtraction has given birth to a baby factoring algorithm.

Download Program (25KB)


Miller-Rabin Primality Test: Demo with source code (beta)

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.)

Miller-Rabin Primality Test

Download Project (10KB) Download Program (17KB)


FermaticTest: Prime and Pseudoprime Calculator

"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.

FermaticTest

Download (34KB)


QTest: Quadratic Sequence Calculator

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.)

QTest

Download (35KB) How to Use QTest


RadiusTest: Polynomial Line and Curve Calculator

Calculate prime and composite distribution in the Sacks Number Spiral by offset (curved series) and angle (straight series).

RadiusTest

Download (22KB)


ZetaTest: Infinite Series Calculator

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.

ZetaTest
To calculate pi, use the screenshot settings

Download (23KB)


PrimeTest: Prime and Composite Analyzer

Prime and composite number calculator (with prime and nonprime factor analyzer and prime number generator).

PrimeTest

Download (31KB)

Tools that are digitally signed are published by: Wordwise Solutions

This site is dedicated to exploring
the
undesigned 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.

 

 

 

 

Recommended sites:

Number Spiral

Number Spirals

Divisor Plot

Interesting Graphing Applets

 © 2007-2013 Michael M. Ross