On the Form of Odd Perfect Gaussian Integers
نویسنده
چکیده
Let Z[i] = {a + bi : a, b ∈ Z} be the ring of Gaussian integers. All Gaussian integers will be represented by Greek letters and rational integers by ordinary Latin letters. Primes will be denoted by π and p respectively. Units will be denoted by ε = ±1,±i and 1 respectively. In 1961 Robert Spira [3] defined the sum-of-divisors function on Z[i] as follows. Let η = εΠπi i be a Gaussian integer. This representation is unique in that we will choose our unit ε such that each πi is in the first quadrant (Re(πi) > 0 and Im(πi) ≥ 0). Spira defined the sum-of-divisors function σ as
منابع مشابه
# A 6 INTEGERS 12 A ( 2012 ) : John Selfridge Memorial Issue ON ODD PERFECT NUMBERS AND EVEN 3 - PERFECT NUMBERS
An idea used in the characterization of even perfect numbers is used, first, to derive new necessary conditions for the existence of an odd perfect number and, second, to show that there are no even 3-perfect numbers of the form 2aM , where M is odd and squarefree and a ≤ 718, besides the six known examples. –In memory of John Selfridge
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