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 that mathematicians think they can solve, but they need more work to show it’s really correct. Many important conjectures have guided mathematicians for years, leading them to discover new ways to solve problems and understand numbers better.

One famous example is the Riemann hypothesis, a conjecture about the distribution of prime numbers. Another well-known one is Fermat's conjecture, which was an unsolved problem for a long time until it was finally 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 their efforts. They push the field forward, leading to new discoveries and deeper understanding. Even when a conjecture turns out to be false, the journey to test it often results in valuable knowledge and new mathematical 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 if many examples support a conjecture, it isn’t considered 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 conjecture is called a hypothesis when it is used often in proving other results. For example, the Riemann hypothesis is a conjecture from number theory that helps predict how prime numbers are distributed. Many experts believe it is true, and some have created proofs that depend on it being true. These are called conditional proofs. However, if the hypothesis turns out to be false, these proofs would not be valid, so there is much interest in proving or disproving such conjectures.

Important examples

Fermat's Last Theorem

Main article: Fermat's Last Theorem

Fermat's Last Theorem is a famous math problem that says you can't find whole numbers that fit a special equation after a certain point. A mathematician named Pierre de Fermat first thought about this in 1637, but it took until 1995 for Andrew Wiles to finally 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 so that no two next-to-each-other areas share the same color. This idea was first noticed in 1852, and 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 about whether two ways of breaking a space into small pieces can always be made to match up. This idea came about in 1908, and while some parts were proven true, 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, proposed by André Weil, helped connect different parts of math, and 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, made by Henri Poincaré in 1904, 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 about where certain special numbers appear. Proposed by Bernhard Riemann, it connects to understanding prime numbers — the building blocks of all numbers. Many mathematicians think it's true, and 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, first clearly stated in 1971 by Stephen Cook, 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 introduced the term "conjecture" in the study of scientific philosophy. In science, a conjecture is similar to a hypothesis, which is a conjecture that can be tested.