Interactive Modular Lifting Rings

Built by Wessen Getachew | Created by Leonhard Euler
φ(n) = n∏(1 - 1/p) | gcd(a,b) = 1 | Z/nZ ≅ (Z/nZ)* | r ≡ s (mod m)

Gap Analysis

Gap 2 (Twin Primes)
Gap 4 (Cousin Primes)
Gap 6 (Sexy Primes)
Gap 8
Gap 10
Gap 12
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Modular Ring System

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No limits - explore any modulus range you want!

Visual Customization

Advanced Animation Engine

Enable Individual Ring Rotation
Reverse Rotation Direction

Lifting Morphisms

Direct Lifts (r → r)
Modular Lifts (r → r + M×2ⁿ)

Display Controls

Show Residue Labels
Invert Ring Nesting Order
Highlight Unit Circle

Prime Sieve Engine

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Advanced Prime Analysis

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Mathematical Analysis

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Mathematical Legend & Notation

Unit Circle (M=1) - Leonhard Euler's foundation
gcd(r,M) = 1 - Euler's totient function φ(M)
Prime gap connections within Euler's rings
Direct lifts: r ↦ r (Euler homomorphism)
Skip-level lifts: r ↦ r (extended morphism)
Modular lifts: r ↦ r + M×2ⁿ (Chinese Remainder)
Custom moduli explorations
Mₙ = 30 × 2ⁿ series (Euler's multiplicative function)