We present a unified theoretical–computational framework that integrates (Riemann in Monatsberichte der Berliner Akademie 1859:671–680, <CitationRef CitationID="CR1">1859</CitationRef>; Davenport in Multiplicative Number Theory, 3rd edn. Springer, New York, <CitationRef CitationID="CR2">2000</CitationRef>):<OrderedList> <ListItem> <ItemNumber>1.</ItemNumber> <ItemContent> Main Algorithm (Bipartite Binomial) for the deterministic factorization of semi‑prime integers (Iwaniec et al. in Analytic Number Theory. American Mathematical Society, Providence, <CitationRef CitationID="CR3">2004</CitationRef>). </ItemContent> </ListItem> <ListItem> <ItemNumber>2.</ItemNumber> <ItemContent> Theorem <InternalRef RefID="FPar1">1</InternalRef> and its full proof via a purely integer‑based example (Serre in A Course in Arithmetic. Springer, New York, <CitationRef CitationID="CR4">1973</CitationRef>; Edwards in Riemann’s Zeta Function. Academic Press, New York, <CitationRef CitationID="CR5">1974</CitationRef>). </ItemContent> </ListItem> <ListItem> <ItemNumber>3.</ItemNumber> <ItemContent> Supplementary Algorithms<UnorderedList Mark="None"> <ItemContent> Algorithm 2: Digit‑count propositions for selecting the correct binomial shift (Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Springer, New York, <CitationRef CitationID="CR6">1990</CitationRef>; Serre in Linear Representations of Finite Groups. Springer, New York, <CitationRef CitationID="CR7">1977</CitationRef>). </ItemContent> <ItemContent> Algorithm 3: Exponentiated arithmetic progression with an automatic stop criterion. </ItemContent> </UnorderedList> </ItemContent> </ListItem> <ListItem> <ItemNumber>4.</ItemNumber> <ItemContent> Auxiliary Propositions establishing bounds and error estimates (Bombieri in Problems of the Millennium: The Riemann Hypothesis. Clay Mathematics Institute. Dis-ponívelem. <CitationRef CitationID="CR8">2000</CitationRef>). </ItemContent> </ListItem> <ListItem> <ItemNumber>5.</ItemNumber> <ItemContent> Corollaries providing modular‑congruence checks for incremental validation (Lagarias in J. Number Theory 39(2):341–438, <CitationRef CitationID="CR9">2000</CitationRef>). </ItemContent> </ListItem> <ListItem> <ItemNumber>6.</ItemNumber> <ItemContent> Proposed Exercises accompanied by detailed solutions (Titchmarsh in The Theory of the Riemann Zeta-Function (2nd ed., ed. D. R. Heath-Brown). Oxford: Oxford University Press, <CitationRef CitationID="CR10">1986</CitationRef>). </ItemContent> </ListItem> <ListItem> <ItemNumber>7.</ItemNumber> <ItemContent> Theorem 3 (Tripartite Binomial)—an optimized extension guaranteeing polynomial‑time behavior (class P) (Conrey in Notices of the AMS 50(3):341–353, <CitationRef CitationID="CR11">2003</CitationRef>). </ItemContent> </ListItem> <ListItem> <ItemNumber>8.</ItemNumber> <ItemContent> Experimental Data and Complexity Analysis confirming the method’s practical performance (Odlyzko in Math. Comput. 48(177):273–308, <CitationRef CitationID="CR12">1987</CitationRef>; Rubinstein in Exp. Math. 3(3):173–197, <CitationRef CitationID="CR13">1994</CitationRef>). </ItemContent> </ListItem> </OrderedList>This framework suggests a deterministic, polynomial‑time approach to semi‑prime factorization and highlights deep connections with the Riemann Hypothesis (Riemann in Monatsberichte der Berliner Akademie 1859:671–680, <CitationRef CitationID="CR1">1859</CitationRef>; Edwards in Riemann’s Zeta Function. Academic Press, New York, <CitationRef CitationID="CR5">1974</CitationRef>; Cramer in Acta Arith. 2:23–46, <CitationRef CitationID="CR14">1937</CitationRef>).

Semiprime Factorization in Style (RSA) is in Class P

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2025年8月19日

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Fabiano Ferreira da Silva

摘要

We present a unified theoretical–computational framework that integrates (Riemann in Monatsberichte der Berliner Akademie 1859:671–680, 1859; Davenport in Multiplicative Number Theory, 3rd edn. Springer, New York, 2000): 1.

Main Algorithm (Bipartite Binomial) for the deterministic factorization of semi‑prime integers (Iwaniec et al. in Analytic Number Theory. American Mathematical Society, Providence, 2004).

Theorem 1 and its full proof via a purely integer‑based example (Serre in A Course in Arithmetic. Springer, New York, 1973; Edwards in Riemann’s Zeta Function. Academic Press, New York, 1974).

Supplementary Algorithms

Algorithm 2: Digit‑count propositions for selecting the correct binomial shift (Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Springer, New York, 1990; Serre in Linear Representations of Finite Groups. Springer, New York, 1977).

Algorithm 3: Exponentiated arithmetic progression with an automatic stop criterion.

Auxiliary Propositions establishing bounds and error estimates (Bombieri in Problems of the Millennium: The Riemann Hypothesis. Clay Mathematics Institute. Dis-ponívelem. 2000).

Corollaries providing modular‑congruence checks for incremental validation (Lagarias in J. Number Theory 39(2):341–438, 2000).

Proposed Exercises accompanied by detailed solutions (Titchmarsh in The Theory of the Riemann Zeta-Function (2nd ed., ed. D. R. Heath-Brown). Oxford: Oxford University Press, 1986).

Theorem 3 (Tripartite Binomial)—an optimized extension guaranteeing polynomial‑time behavior (class P) (Conrey in Notices of the AMS 50(3):341–353, 2003).

Experimental Data and Complexity Analysis confirming the method’s practical performance (Odlyzko in Math. Comput. 48(177):273–308, 1987; Rubinstein in Exp. Math. 3(3):173–197, 1994).

This framework suggests a deterministic, polynomial‑time approach to semi‑prime factorization and highlights deep connections with the Riemann Hypothesis (Riemann in Monatsberichte der Berliner Akademie 1859:671–680, 1859; Edwards in Riemann’s Zeta Function. Academic Press, New York, 1974; Cramer in Acta Arith. 2:23–46, 1937).