## Can conjectures always be proven true?

A conjecture is a mathematical statement that has not yet been rigorously proved. Conjectures arise when one notices a pattern that holds true for many cases. However, just because a pattern holds true for many cases does not mean that the pattern will hold true for all cases.

**How can you prove that a conjecture is false?**

To show that a conjecture is false, you have to find only one example in which the conjecture is not true. This case is called a counterexample. To show that a conjecture is always true, you must prove it. A counterexample can be a drawing, a statement, or a number.

### What is the difference between conjecture and hypothesis?

The word conjecture is defined as an opinion based on incomplete information. Conjecture is an idea, hypothesis is a conjecture that can be tested by experiment or observation, and consensus emerges when other interested colleagues agree that evidence supports a hypothesis that has explanatory value.

**Are postulates accepted without proof?**

Postulates are accepted as true without proof. A logical argument in which each statement you make is supported by a statement that is accepted as true. In a conditional statement, the statement that immediately follows the word if.

## Does a theorem become a definition after proven true?

Theoremhood and truth All theorems were proved by using implicitly or explicitly these basic properties, and, because of the evidence of these basic properties, a proved theorem was considered as a definitive truth, unless there was an error in the proof.

**What is formal proof method?**

In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference.

### What is the Weil conjecture in math?

Weil conjectures. In mathematics, the Weil conjectures were some highly influential proposals by André Weil (1949), which led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.

**Is there a Weil cohomology for complex varieties?**

Weil suggested that the conjectures would follow from the existence of a suitable ” Weil cohomology theory ” for varieties over finite fields, similar to the usual cohomology with rational coefficients for complex varieties.

## Who proved the rationality of the conjectures?

The rationality part of the conjectures was proved first by Bernard Dwork ( 1960 ), using p -adic methods.

**What is the difference between topology and Weil’s Weil theory?**

Given that finite fields are discrete in nature, and topology speaks only about the continuous, the detailed formulation of Weil (based on working out some examples) was striking and novel. It suggested that geometry over finite fields should fit into well-known patterns relating to Betti numbers, the Lefschetz fixed-point theorem and so on.