A Discussion with Gemini AI About Mathematical Conjectures

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I asked Google Gemini AI about mathematical conjectures. Specifically, those beyond Collatz, Goldbach, and the Reiman hypothesis.

Question: What mathematical conjecture is as famous as the Collatz and Goldbach conjectures?

Response: There's a strong contender for a conjecture as famous (or even more famous) than Collatz and Goldbach: The Riemann Hypothesis.

Question: Yes, what would be the next three top conjectures?
Response: Defining the "top" conjectures can be subjective, but here are three other highly influential and challenging conjectures that mathematicians are actively trying to solve:

1. Birch and Swinnerton-Dyer Conjecture: This conjecture bridges number theory and algebraic geometry. It relates the arithmetic properties of an elliptic curve (a specific type of equation) to its geometric behavior. Proving it would have significant implications for understanding both areas of mathematics.

2. P vs. NP Problem: This problem sits at the heart of computer science and complexity theory. It asks if every problem whose solution can be quickly verified can also be solved quickly. A solution to P vs. NP would have major ramifications for cryptography, optimization problems, and various computational fields.

3. Hodge Conjecture: This conjecture connects complex geometry and algebraic topology. It proposes a deep relationship between certain geometric objects (harmonic forms) and topological properties of complex manifolds (higher-dimensional spaces). Proving it would unify these areas of mathematics and lead to new insights into complex geometric structures.

These conjectures represent diverse areas of mathematics and highlight the range of unsolved problems mathematicians grapple with.

Best Mathematical Equations

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I believe most people would give top marks to Euler's Identity as the most beautiful mathematical equation as it ties into one, simple equation, five key mathematical constants (unity, Pi, zero, e, and i). Recently, Scientific America, published an article listing several more beautiful mathematical equations: https://www.scientificamerican.com/article/these-are-the-most-beautiful-equations-in-mathematics/. Of these, I like Euler's Product as a very close second place.

Update: July 6, 2024: IFL Science recently published a list of the three most important equations: https://www.iflscience.com/three-incredible-equations-and-why-theyre-so-important-74961

Cicadas - Masters of prime numbers and metallurgy

In a prior post I noted how cicadas use a prime number of years to maximize their survival against predators  ( https://www.blogger.com/blog/post/edit/8711601538345502519/2640846713843380019).  It also turns our that the females use a body part, a depositor, composed of manganese, zinc, and iron to deposit her eggs. Therefore, cicadas are infests' masters of prime numbers and metallurgy.

See video at: https://www.youtube.com/watch?v=0y5yTNdVb68.