Is 2P1 a perfect number?
Theorem 1 (Euclid’s Perfect Number Theorem). If 2p1 is a prime number, then 2p1(2 1) is a perfect number. Proof. Let pbe an integer such that 2p1 is a prime number. We aim to show that 2p1(2 1) is a perfect number. Now let q= 2p1, so that 2p 1(2 1) = 2p 1q.
What is a perfect number in math?
Perfect number, a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3. Other perfect numbers are 28, 496, and 8,128.
Is 2n a perfect number?
THEOREM 8.9 (Euclid) If n is an integer ? 2 such that 2n ? 1 is a prime, then N = 2n?1 (2n ? 1) is a perfect number. PROOF Since 2n ? 1 is a prime, ? (2n ? 1) = 1 + (2n ? 1) = 2n .
What is the odd perfect number theorem?
By Euler’s Odd Perfect Number Theorem, all odd perfect numbers are congruent to 1 (mod 4), so either n1 (mod 6) and n1 (mod 4) or n3 (mod 6) and n1 (mod 4). Using these two relations, we can reason that nmust be of the form 12m+1 or 12m+9.