Tool and algorithm to disprove the Polya conjecture. The Polya conjecture proposes that the majority of values of the prime factor count among numbers less than a specific integer are odd.
Pólya Conjecture - dCode
Tag(s) : Arithmetics, Algorithm
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Pólya Conjecture
Statement of the Conjecture
In number theory, the Polya conjecture, proposed by the Hungarian mathematician George Polya in 1919, states: For any integer $ N > 1 $, the set of prime factorization of positive natural numbers less than $ N $ consists of more factorizations with an odd number of factors than of factorizations with an even number of factors.
This conjecture is false, the first counterexample is $ N = 906150257 $. Since the smallest counterexample is quite large, it is often used to show that a conjecture can be verified several million times while still being false.
Answers to Questions (FAQ)
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 $.
The number 1 has no prime factors, so 0 factor, its decomposition is considered even.
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; // number 1
var odd = 0;
var factors = [];
for (i = 2; i < 1000000000; i++) {
factors = prime_factors(i); // return [factor1, factor2, ...]
if (factors.length % 2) odd++;
else even++;
if (even > odd) {
alert(i);
break;
}
}
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