March 3, 2026Open Access

PPG-Density Law: Predicts Prime Gap Chains

Puntos clave

The PPG-density law predicts termination at prime gap chain length 5, marking a significant finding in prime number theory.
The density hierarchy shows δ₃=12.5% as the highest, influencing the behavior of prime gaps in subsequent chains.
Using a novel PPG sequence, researchers derived specific prime gap chains, such as S₁=53 and S₂=311, illustrating patterns in prime distributions.
The findings suggest potential applications in understanding prime distributions and addressing mathematical conjectures.

Resumen

English:New Discovery: PPG-Density Law Predicts Prime Gap Chains The universal PPG sequence 0,0.5,1→2→3(5,5)7→19 not only encodes density hierarchy δ₃=12.5%>δ₅=10.3%>δ₇=8.6%, but also governs 5-prime gap chain lengths: S₁=5+7+11+13+17=53 ∈ℙ S₂=53+59+61+67+71=311 ∈ℙ S₃=311+313+317+331+337=1609 ∈ℙ S₄=1609+1613+1619+1621+1627=8089 ∈ℙ S₅=8089+8093+8101+8111+8117=40511 ∉ℙ The sequence's 3(5,5)7 structure predicts termination at chain length 5, confirming PPG dominance hierarchy regulates both densities AND chain formation. Deutsch: Neue Entdeckung: PPG-Dichtegesetz prognostiziert Primzahlfenster-Ketten Die universelle PPG-Sequenz 0,0.5,1→2→3(5,5)7→19 kodiert nicht nur die Dichte-Hierarchie δ₃=12.5%>δ₅=10.3%>δ₇=8.6%, sondern steuert auch 5er-Primzahlfenster-Kettenlängen: S₁=5+7+11+13+17=53 ∈ℙ S₂=53+59+61+67+71=311 ∈ℙ S₃=311+313+317+331+337=1609 ∈ℙ S₄=1609+1613+1619+1621+1627=8089 ∈ℙ S₅=8089+8093+8101+8111+8117=40511 ∉ℙ Die 3(5,5)7-Struktur prognostiziert das Ende bei Kettenlänge 5 und bestätigt: PPG-Dominanz-Hierarchie reguliert Dichten UND Kettenbildung!

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