Do you plan on covering Thue’s Lemma at some point? There is an efficient algorithm for the result it gives which is rather unexpected, and that allows one to decompose 4n+1 primes into a sum of two squares relatively quickly. As an example of the algorithm, 1009 is 4*(252)+1. One can then find (+-469)^2 = -1(mod 1009) by using exponentiation by squaring mod 1009 and possibly using quadratic reciprocity to find a quadratic nonresidue to raise to the (p-1)/4 -th power. Now the algorithm given by Thue’s Lemma starts running the Euclidean algorithm on p = 1009 and the square root of -1 that was found (469) until the last two remainders drop below the square root of p = 1009, (31 < sqrt(1009) < 32): 1009 469 71 43 28 (
Professor, are you planning to cover about Dirichlet class number formula for binary quadratic forms?
Do you plan on covering Thue’s Lemma at some point? There is an efficient algorithm for the result it gives which is rather unexpected, and that allows one to decompose 4n+1 primes into a sum of two squares relatively quickly.
As an example of the algorithm, 1009 is 4*(252)+1. One can then find (+-469)^2 = -1(mod 1009) by using exponentiation by squaring mod 1009 and possibly using quadratic reciprocity to find a quadratic nonresidue to raise to the (p-1)/4 -th power. Now the algorithm given by Thue’s Lemma starts running the Euclidean algorithm on p = 1009 and the square root of -1 that was found (469) until the last two remainders drop below the square root of p = 1009, (31 < sqrt(1009) < 32):
1009
469
71
43
28 (
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