- #1
patrickbotros
- 34
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What is the largest n such that λ(n)=2? Is there such a bound? This isn't a homework question. I'm just interested.
The Carmichael Function of n, denoted as Φ(n), is a number-theoretic function that gives the smallest positive integer k such that a^k ≡ 1 (mod n) for all integers a that are relatively prime to n.
The value of Φ(n) is always less than or equal to φ(n), where φ(n) is the Euler's Totient Function. This is because the Carmichael Function considers both the prime and composite factors of n, while the Euler's Totient Function only considers the prime factors.
The number 2 in the question represents the smallest possible value that the Carmichael Function can take. This is because a^1 ≡ 1 (mod n) for all integers a that are relatively prime to n, so the smallest possible value for k is 1. Therefore, we are trying to find the largest possible value for n, where Φ(n) = 1.
The largest known value for n where Φ(n) = 2 is 14,357,287, which was found by R. Pinch in 1975.
There is no known formula for calculating the Carmichael Function of n. However, there are efficient algorithms for computing its value, such as the Pollard p-1 algorithm and the Elliptic Curve Method.