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Pólya Conjecture

Tool (algorithm) to invalidate the Polya conjecture. Polya's conjecture suggests that the majority of the prime factor numbers of numbers less than a precise integer is odd.

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Pólya Conjecture -

Tag(s) : Arithmetics, Algorithm

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Pólya Conjecture

Statement of the Conjecture

In number theory, Pólya's conjecture, proposed by the Hungarian mathematician George Pólya in 1919, states that for any integer $ N $, in the decomposition into prime factors of all natural integers less than $ N $, there are more decompositions with an odd number of factors than decompositions with an even number of factors.

This conjecture is false, the first counterexample is $ N = 906150257 $

Tool (algorithm) to invalidate the Polya conjecture. Polya's conjecture suggests that the majority of the prime factor numbers of numbers less than a precise integer is odd.

Answers to Questions

How to prove the Polya conjecture?

To prove that a conjecture is true, a rigorous mathematical proof is needed. To prove that the conjecture is false, it is enough to give one counter-example.

Example: For $ N = 10 $, there are 5 decompositions with an odd number of factors: $ 8, 7, 5, 3, 2 $, and 4 decompositions with an even number of factors: $ 9, 6, 4, 1 $. Since $ 5 > 4 $, the conjecture is true for $ N = 10 $, but this does not mean that it is true for all $ N $.

What is the first counterexample?

The Polya conjecture was refuted in 1958, so it is false. The smallest counterexample is the number $ 906150257 $.

What is the Polya check algorithm?

The algorithm corresponding to the verification of the conjecture is similar to the following:// Javascript
var even = 1;
var odd = 0;
var d = new Array();
for (i = 2; i < 4000000000; i++) {
d = prime_factor_decomposition(i); // return a table with all factors
if (d.length % 2) odd++;
else even++;
if (even > odd) {
alert(i);
break;
}

Source code

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