Python scripts for figures in:

"An elementary heuristic for Hardy–Littlewood extended Goldbach's conjecture" DOI:10.1007/s40863-019-00146-3 / arXiv:1508.05702

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Description

Fixed files

• generators/*gen.py: Generators of lists (phi(n), primes and r_{P,2}(n), described below).

• generators/figure_*.py: Generators of images (described below)

• params.txt: Tells generators the upper limit of each list or image (default: primes up to 10^7 [primegen.py], Goldbach's comet up to 10^6 [figure_2.py], values of phi(n) up to 10^5 [phigen.py and figure1.py])

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Generated files

Images

• imgs/figure_1.png: Comparison between the graphs of phi(n) and pi(n)-omega(n), for even n <= 10^5.

• imgs/figure_2.png: Graph of the function r_{P,2}(n), which counts the in how many ways n can be written as the sum of two primes (counting repetition), for even n <= 10^6.

• imgs/figure_3.png: Graph of the ratio between r_{P,2}(n) and Hardy--Littlewood's conjectured estimate, for even n <= 10^7.

Lists

• lists/phis.txt: Values of Euler's totient function phi(n) for n <= 10^5, separated by space.

• lists/primes.txt: List of primes <= 10^7, separated by space.

• lists/r2ns.txt: Values of r_{P,2}(n) for n <= 10^7, separated by space.

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How to run

Linux

1. Run gen_lists.sh
2. Run gen_imgs.sh

Not-linux

1. Run generators/phigen.py
2. Run generators/primegen.py
3. Run generators/r2gen.py
4. Run generators/figure_i.py (i = 1,2,3)

In both cases, lists are stored in ~/lists and images are stored in ~/imgs.

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Dependencies

All the Python scripts were written for Python 2.7. Apart from standard Python libraries, SciPy and Matplotlib are necessary to run the scripts.

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Time estimate

All programs run pretty fast (less than 30 secs), with the notable exception of figure_3.py. For a basis of comparison, a laptop with the specs

• Processor: Intel(R) Core(TM) i3-5005U CPU @ 2.00GHz,
• Memory: 4GiB, clock 1600MHz (0.6ns),
• HDD: WDC WD5000LPCX-6 500GB,

running figure_3.py for n up to 10^7 takes approximately 1h40mins.