Conjecture

In mathematics, a conjecture is an idea or statement that someone believes might be true, but it hasn’t been proven yet. It’s like a puzzle mathematicians want to solve, but they still need more work to show it’s correct.

One famous example is the Riemann hypothesis, a conjecture about prime numbers. Another well-known one is Fermat's conjecture, which was unsolved for a long time until it was proven in 1995 by the mathematician Andrew Wiles. When a conjecture is proven true, it becomes a theorem, meaning it’s now a solid part of mathematics.

Conjectures are important because they help mathematicians know where to focus. They push the field forward, leading to new discoveries and a deeper understanding. Even when a conjecture turns out to be false, testing it often leads to valuable knowledge and new tools.

Resolution of conjectures

In mathematics, a conjecture is an idea that mathematicians think might be true, but they haven’t proven it yet. Even with many examples that support a conjecture, it isn’t proven until someone shows it cannot be false. There are different ways to prove a conjecture, including checking every possible case, sometimes using computers.

When a conjecture is proven, it becomes a theorem. Some important theorems were once conjectures, like Fermat's Last Theorem. Sometimes, a conjecture turns out to be false when a counterexample is found. Other conjectures, like the continuum hypothesis, can’t be proven or disproven using the usual rules of set theory, so they remain independent.

Conditional proofs

Sometimes, a guess is called a hypothesis when it helps prove other results. For example, the Riemann hypothesis is a guess from number theory that helps figure out how prime numbers are spread out. Many experts think it is true, and some have made proofs that work if it is true. These are called conditional proof. But if the hypothesis turns out to be false, these proofs would not be right, so people are very interested in proving or disproving such guesses.

Important examples

Fermat's Last Theorem

Main article: Fermat's Last Theorem

Fermat's Last Theorem is a famous math problem. It says you can't find whole numbers that fit a special equation after a certain point. A mathematician named Pierre de Fermat thought about this in 1637. It took until 1995 for Andrew Wiles to prove it true. This problem helped create new areas of math!

Four color theorem

Main article: Four color theorem

The four color theorem tells us that you only need four colors to color any map. No two next-to-each-other areas will share the same color. This idea was first noticed in 1852. It took until 1976 for Kenneth Appel and Wolfgang Haken to prove it using a computer. Their work showed that no map could ever need more than four colors.

Hauptvermutung

Main article: Hauptvermutung

The Hauptvermutung is a guess in geometry. It is about whether two ways of breaking a space into small pieces can always be made to match up. This idea came about in 1908. Some parts were proven true, but the full idea was later shown to be false.

Weil conjectures

Main article: Weil conjectures

The Weil conjectures are guesses about patterns in solving equations over special number systems. These ideas were proposed by André Weil. They helped connect different parts of math. Over time, different pieces of them were proven true.

Poincaré conjecture

Main article: Poincaré conjecture

The Poincaré conjecture is about whether certain kinds of 3D spaces must always look like a simple ball. This guess was made by Henri Poincaré in 1904. It was finally proven true by Grigori Perelman in 2003. Solving it was one of the biggest open questions in geometry.

Riemann hypothesis

Main article: Riemann hypothesis

The Riemann hypothesis is a big guess in math. It is about where certain special numbers appear. It was proposed by Bernhard Riemann. It connects to understanding prime numbers — the building blocks of all numbers. Many mathematicians think it's true. Solving it is one of the biggest challenges today.

P versus NP problem

Main article: P versus NP problem

The P versus NP problem asks whether every problem that’s easy to check can also be easy to solve. This question was first clearly stated in 1971 by Stephen Cook. It is one of the biggest open problems in computer science. Many believe the answer is no, but no one has proven it yet.

Other conjectures

• Goldbach's conjecture
• The twin prime conjecture
• The Collatz conjecture
• The Manin conjecture
• The Maldacena conjecture
• The Euler conjecture
• The Hardy-Littlewood conjectures
• The Langlands program

In other sciences

Karl Popper used the term "conjecture" in the study of scientific philosophy. In science, a conjecture is like a hypothesis. A hypothesis is a conjecture that we can test.