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Tools for studying and exploring prime numbers, Goldbach partitions, fractals, etc.
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LICENSE
README.md
evens_from_odds.html
goldbach.html
jams.html
jams2.html
jams2.js
jams3.html
jams3.wat
main.py
misc.py
polar.html
prime_gaps.html
primes.py

README.md

primes.py

A script for analyzing Goldbach partitions.

Currently testing the following observation/conjecture (12/31/2019) and its implications:

For every prime P, there exists a threshold N such that every even number > N is the sum of two primes > P.

N is the lowest such threshold, i.e. N itself is not the sum of two primes > P.

N / P goes to 2 as P goes to infinity.

For example, the first few pairs (P, N) are (2, 4), (3, 8), (5, 16), (7, 20), (11, 28), ...

(See, for example, python primes.py maxevens 100000)

This suggests a sliding window approach to generating all even numbers > 2 as sums of two primes by generating subsets of consecutive even numbers from subsets of consecutive primes. (See, for example, python primes.py sliding 50000 which runs such an algorithm to generate all even numbers from 6 to 99,650.)

This resulted from observing correlations between sums of two primes of certain residue classes.

For example:

  1. For any two primes P1 and P2, both 1 mod 4, there exists at least one pair of primes Q1 and Q2, both 3 mod 4, such that P1 + P2 = Q1 + Q2. (P1 can be equal to P2, and Q1 may be equal to Q2.)

    (See, for example, python primes.py modstats 100000 4)

  2. 3 + (any prime that's 1 mod 6) is the sum of two primes that are both 5 mod 6, and

    3 + (any prime that's 5 mod 6 and > 5) is the sum of two primes that are both 1 mod 6.

    This means that every even number > 8 is the sum of two primes > 3.

    (See, for example, python primes.py modstats 100000 6)

Further observations (verified for all evens up to 50 million):

  • Every even number > 68 is the sum of two primes neither of which is 1 mod 10. (python primes.py verify 50000000 1,10) (1/3/2020)

  • Every even number > 68 is the sum of two primes neither of which is 7 mod 10. (python primes.py verify 50000000 7,10) (1/3/2020)

  • Every even number > 152 is the sum of two primes neither of which is 3 mod 10. (python primes.py verify 50000000 3,10) (1/3/2020)

  • Every even number > 152 is the sum of two primes neither of which is 9 mod 10. (python primes.py verify 50000000 9,10) (1/3/2020)

  • Every even number > 2 is the sum of two primes neither of which is 1 mod 8. (python primes.py verify 50000000 1,8) (1/4/2020) This implies that every even number > 2 can be expressed as the sum of two primes using only 3/4 of all primes.

  • Every even number > 56 is the sum of two primes neither of which is 3 mod 8. (python primes.py verify 50000000 3,8) (1/4/2020)

  • Every even number > 188 is the sum of two primes neither of which is 5 mod 8. (python primes.py verify 50000000 5,8) (1/4/2020)

  • Every even number > 188 is the sum of two primes neither of which is 7 mod 8. (python primes.py verify 50000000 7,8) (1/4/2020)

  • Every even number > 2 is the sum of two primes neither of which is 1 mod 7 or 4 mod 7. (python primes.py verify 50000000 4,7,1,7) (1/4/2020) This implies that every even number > 2 can be expressed as the sum of two primes using only 2/3 of all primes.

  • Every even number > 2 is the sum of two primes neither of which is 1, 4, 6, 8, 9, 10, or 15 mod 17. (python primes.py verify 50000000 1,17,4,17,6,17,8,17,9,17,10,17,15,17) (1/4/2020) This omits (3001134 - 1688268) / 3001134 = 43.75% of primes.

goldbach.html

Visualization of patterns in the number of Goldbach partitions.

See https://nightjuggler.com/math/goldbach.html

By default, a 60x60 grid is created. Each square in the grid represents an even number from 6 to 7204 (4 + 2 x 60 x 60). The grid dimensions, the first even number (corresponding to the top-left square), the timeout delay (in milliseconds), and the coloring function (a number from 1 to 5) can be specified in the URL. For example:

https://nightjuggler.com/math/goldbach.html?size=105x40&first=1000&color=3

evens_from_odds.html

Graph of the minimum (*) number of odd numbers > 1 (Y axis) (as calculated by the evens_from_odds function in misc.py) needed to express every even number > 4 up to a threshold (X axis) as the sum of two odds.

See https://nightjuggler.com/math/evens_from_odds.html

The minimum and maximum values of X and whether to connect the data points with a stepped line can be specified in the URL. For example:

https://nightjuggler.com/math/evens_from_odds.html?min=98000&max=102000&line

(*) The evens_from_odds function doesn't always find a minimal set of odd numbers for a given threshold, but it appears to come very close, and it's much faster than evens_from_odds_recursive which can find all sets (of odd numbers) that satisfy the property that every even number > 4 up to the threshold can be expressed as the sum of two numbers from the set.

prime_gaps.html

Zoomable visualization of prime gaps.

See https://nightjuggler.com/math/prime_gaps.html

The maximum number up to which prime gaps will be shown and the modulus used to determine the color of the gaps can be specified in the URL. For example:

https://nightjuggler.com/math/prime_gaps.html?max=50000&mod=5

polar.html

JavaScript for graphing spirals and roses.

See https://nightjuggler.com/math/polar.html

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