A New Divisor Sum Property Unifying Even Perfect Numbers and a Class of Odd Composites.
Year: 2026
Type: article
Abstract: We introduce and study a new property of positive integers defined entirely by the structure of the ordered divisor sequence. For any positive integer that is not a perfect square, the total number of positive divisors is even; write . Let be the complete ordered sequence of . Define the lower divisor sum and the upper divisor sum . We say that satisfies the Bisected Divisor Sum Square Property (BDSSP) if . This condition is equivalent to where denotes the standard sum of divisors function and implies that any integer satisfying the BDSSP while also being perfect must be a triangular number of the form . We prove three theorems. Theorem establishes that every even perfect number satisfies the BDSSP. For with a Mersenne prime, the lower divisors are the consecutive powers which sum to ; the upper divisors are those same powers scaled by the Mersenne prime giving . Theorem proves that for any prime and positive integer such that is also prime, the integer satisfies the BDSSP. The proofs of Theorems and share a single algebraic mechanism: the lower half of the divisor sequence is a finite geometric sum equal to a prime and the upper half is times that same sum yielding . Even perfect numbers and nine of the fourteen odd composites identified computationally are explicit instances of this mechanism unified under Theorem Theorem proves that among all integers of the Euclid form , the condition holds if and only if is an even perfect number, providing a complete characterization of even perfect numbers within the Euclid family via the divisor count function alone. A complete computational search over all integers in the interval verified by independent Python computation and requiring seconds of total running time, identifies exactly fourteen odd composites and seven even integers satisfying the BDSSP. The seven even integers include the five known even perfect numbers and together with two even non-perfect integers and , both of which share the value and the same sum of divisors . We confirm by direct computation that none of the fourteen odd composites satisfies so none is a perfect number. No integer up to satisfies the analogous cubic condition . Checking the fourteen odd composites against the theorem of Roberts , which states that any odd perfect number must satisfy or , we find that nine of the fourteen satisfy none of these three congruences and are therefore eliminated as odd perfect number candidates by the contrapositive of Roberts’s theorem, independently of any computation of . We further confirm that none of the fourteen is divisible by , consistent with the theorem of Sylvester . Neither the sequence of fourteen odd composites nor the sequence of seven even integers appears in the Online Encyclopedia of Integer Sequences and to the best of our knowledge, this class of integers has not been previously studied. We establish no connection between the BDSSP and the existence or non-existence of odd perfect numbers. The property is not proved to be a necessary condition for odd perfect numbers and we do not claim that any odd composite satisfying the BDSSP is related to such numbers. Whether every odd perfect number, if any exists, satisfies the BDSSP is posed as an open problem. If an odd perfect number does satisfy the BDSSP, we derive that must be a triangular number with adding one more entry to the known web of necessary conditions for odd perfect numbers. We conjecture that there are infinitely many odd composites satisfying the BDSSP and infinitely many even integers satisfying the BDSSP. The five odd composites not accounted for by Theorem are identified explicitly and their structure is posed as an open problem. As illustrations of the scalability of the BDSSP framework, we present Euclid form candidates satisfying the simplified form of the BDSSP at the scale of one hundred million digits and one billion digits, with the first and last twenty digits of the Mersenne computed exactly via modular and logarithmic arithmetic. The primality of the Mersenne component in each large-scale candidate has not been tested. These candidates are presented as demonstrations of the analytical reach of the BDSSP and not as claims of new Mersenne prime or even perfect number discoveries.
Author: Mututu, Christoper Muoki
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222711462
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