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.

Pólya Conjecture - dCode

Tag(s) : Mathematics, Algorithm

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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.

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.

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

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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polya,prime,factor,decomposition,odd,even,906150257

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