Unit Circle Visualization
Current Analysis
Adjust parameters to see how major and minor arcs change. The visualization shows the unit circle with major arcs in blue and minor arcs in red.
Progress toward 2n^(1/2+ε): 25%
Minor Arc Methods
Current Techniques:
• Vinogradov's Method: Classical exponential sum estimates give 2n^(3/4+ε)
• Van der Corput: Iterative techniques for oscillatory integrals
• Bombieri-Iwaniec: Advanced exponential sum bounds using spacing
Target Methods for 2n^(1/2+ε):
• Efficient Congruencing: Wooley's breakthrough method
• Decoupling Theory: Bourgain-Demeter-Guth techniques
• Restriction Theory: Harmonic analysis approach
• Mean Value Theorems: Improved bounds via nested congruencing
Additional Information
The key bottleneck is achieving bounds better than 2n^(3/4+ε) on minor arcs. Current methods fall short due to fundamental limitations in exponential sum estimates.
Goldbach's conjecture posits that every even natural number greater than 2 can be expressed as the sum of two primes. Click here to find a Goldbach pair for n, where n is an even natural number >2.
Parameters
Minor Arc Methods
Visualization
Optimization Strategy
Current Status: Seeking breakthrough
Key Challenges:
• Type I/II sum separation in efficient congruencing
• Optimizing congruence classes and weights
• Managing iteration depth vs. polynomial degree
• Balancing major arc coverage with minor arc bounds
Promising Directions:
• Hybrid decoupling + efficient congruencing
• Non-uniform estimates via restriction theory
• Optimized Weyl differencing in nested applications