Lesson plan
FRACTRAN
FRACTRAN is a Turing-complete esoteric programming language invented by the mathematician John Conway. A FRACTRAN program is an ordered list of positive fractions together with an initial positive integer input n. The program is run by updating the integer n as follows: #for the first fraction f in the list for which nf is an integer, replace n by nf #repeat this rule until no fraction in the list produces an integer when multiplied by n, then halt. gives the following FRACTRAN program, called PRIMEGAME, which finds successive prime numbers: \left( \frac{17}{91}, \frac{78}{85}, \frac{19}{51}, \frac{23}{38}, \frac{29}{33}, \frac{77}{29}, \frac{95}{23}, \frac{77}{19}, \frac{1}{17}, \frac{11}{13}, \frac{13}{11}, \frac{15}{2}, \frac{1}{7}, \frac{55}{1} \right) Starting with n=2, this FRACTRAN program generates the following sequence of integers: * 2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, ... , i.e. the sequence of PRIMEGAME numbers After 2, this sequence contains the following powers of 2: 2^2=4,\, 2^3=8,\, 2^5=32,\, 2^7=128,\, 2^{11}=2048,\, 2^{13}=8192,\, 2^{17}=131072,\, 2^{19}=524288,\, \dots The exponent part of these powers of two are primes, 2, 3, 5, etc.
Article summary
FRACTRAN is a Turing-complete esoteric programming language invented by the mathematician John Conway. A FRACTRAN program is an ordered list of positive fractions together with an initial positive integer input n. The program is run by updating the integer n as follows: #for the first fraction f in the list for which nf is an integer, replace n by nf #repeat this rule until no fraction in the list produces an integer when multiplied by n, then halt. gives the following FRACTRAN program, called PRIMEGAME, which finds successive prime numbers: \left( \frac{17}{91}, \frac{78}{85}, \frac{19}{51}, \frac{23}{38}, \frac{29}{33}, \frac{77}{29}, \frac{95}{23}, \frac{77}{19}, \frac{1}{17}, \frac{11}{13}, \frac{13}{11}, \frac{15}{2}, \frac{1}{7}, \frac{55}{1} \right) Starting with n=2, this FRACTRAN program generates the following sequence of integers: * 2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, ... , i.e. the sequence of PRIMEGAME numbers After 2, this sequence contains the following powers of 2: 2^2=4,\, 2^3=8,\, 2^5=32,\, 2^7=128,\, 2^{11}=2048,\, 2^{13}=8192,\, 2^{17}=131072,\, 2^{19}=524288,\, \dots The exponent part of these powers of two are primes, 2, 3, 5, etc.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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