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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) : Mathematics, Algorithm

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Pólya Conjecture

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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, taking the decomposition into prime factors of all natural integers less than N, then 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, one example is sufficient.

Example: 'N = 10', there are 5 decompositions with an odd number of factors: 8, 7, 5, 3 and 2, and 4 decompositions with an even number of factors: , 6 , 4 and 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 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;
}

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Source code

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