Liouville Function
A completely multiplicative function in number theory, defined for a positive integer n as (-1)^Ω(n), where Ω(n) denotes the number of prime factors of n counted with multiplicity. The function takes the value −1 when n has an odd number of prime factors and +1 when n has an even number of prime factors.
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The classical Goldbach conjecture requires decomposing an even number into the sum of two primes, whereas the Liouville weakened version only requires that both addends have an odd total number of prime factors (Liouville function value of -1), allowing composite numbers to qualify.
The prime factorization of 12 is 2×2×3, with a total of three prime factors (an odd number), so its Liouville value is -1.
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The classical Goldbach conjecture requires decomposing an even number into the sum of two primes, whereas the Liouville weakened version only requires that both addends have an odd total number of prime factors (Liouville function value of -1), allowing composite numbers to qualify.
50%UnverifiedThe prime factorization of 12 is 2×2×3, with a total of three prime factors (an odd number), so its Liouville value is -1.
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