Every Nontrivial Finite MRB Partial Sum Is Irrational Lebsir Version
Marvin Ray Burns
Marvin Ray Burns, Department of Math, Indiana University of Indianapolis, United States of America (USA).
Manuscript received on 24 August 2026 | First Revised Manuscript received on 31 August 2026 | Second Revised Manuscript received on 21 September 2026 | Manuscript Accepted on 15 October 2026 | Manuscript published on 30 October 2026 | PP: 51-55 | Volume-6 Issue-2, October 2026 | Retrieval Number: 100.1/ijam.B125406021026 | DOI: 10.54105/ijam.B1254.06021026
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Abstract: Let SN= ∑ (-1)Nnn=1(n1/n-1) denote the Nth partial sum of the defining series for the MRB constant. We prove that every nontrivial finite partial sum is an irrational algebraic number: SN ∈ Q (n>2). The proof is entirely finite and algebraic. The proof uses the vanishing field trace of each radical n1/n, together with the exact contribution of the rational terms in the alternating sum. We also interpret the fractional-part representation SN= ∑ (-1)Nnn=1 Fractional Part (n1/n) within Habib Lebsir’s congruence framework modulo 1/n. The MRB sequence corresponds to the diagonal choice x=n in Lebsir’s family of partitions, for which [n1/n] = 1 (n>2).Thus, rationality never occurs at a finite stage of the defining MRB process. This does not determine the arithmetic nature of the infinite MRB constant. Still, it shows that any rationality of the limit, were it to occur, could arise only through the infinite limiting process and not through a finite algebraic cancellation among its defining radicals.
Keywords: MRB Constant, Algebraic Irrationality, Field Trace, Radical Extensions, Fractional Parts, Lebsir Congruences.
Scope of the Article: Number Theory