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) : Arithmetics, Algorithm

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!

A suggestion ? a feedback ? a bug ? an idea ? *Write to dCode*!

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.

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 $.

The **Polya 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;

}

dCode retains ownership of the source code of the script Pólya Conjecture online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Pólya Conjecture script for offline use on PC, iPhone or Android, ask for price quote on contact page !

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

Source : https://www.dcode.fr/polya-conjecture

© 2020 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.

Feedback

▲