Distinctive Style of Mathematical Papers

Mathematical Statements

The object of a typical paper is to establish mathematical statements. mathematical statements are the main currency of mathematics.

A conjecture is a mathematical statement that the author firmly believes but cannot prove.

A mathematical statement is established by means of a proof.

Theorems

The most important of these statements are usually called theorems.

The statements that are the main goals of a paper are usually called theorems.

A theorem is a statement that you regard as intrinsically interesting, a statement that you might think of isolating from the paper and telling other mathematicians about in a seminar, for instance. The statements that are the main goals of a paper are usually called theorems.

The main aim of an article in mathematics is usually to prove theorems.

Propositions

A proposition is a bit like a theorem, but it tends to be slightly “boring.”

Lemmas

Lemma 1.1.15 Let A be a square matrix that has a right inverse, a matrix R such that AR = I and also a left inverse, a matrix L such that LA = I. Then R = L. So A is invertible and R is its inverse.

Proof R = IR = (LA)R = L(AR) = LI= L.

– Chapter 1 Matrices, Algebra, by Michael Artin

Often, if you are trying to prove a theorem, the proof becomes long and complicated, in which case if you want anybody to read it you need to make the structure of the argument as clear as possible. One of the best ways of doing this is to identify subgoals, which take the form of statements intermediate between your initial assumptions and the conclusion you wish to draw from them. These statements are usually called lemmas.

One can draw a parallel with computer programming: if you are writing a complicated program, it is good practice to divide your main task into subtasks and write separate mini-programs for them, which you can then treat as “black boxes,” to be called upon by other parts of the program whenever they are useful.

Some lemmas are difficult to prove and are useful in many different contexts, so the most important lemmas can be more important than the least important theorems. However, a general rule is that a result will be called a lemma if the main reason for proving it is in order to use it as a stepping stone toward the proofs of other results.

Corollaries

A corollary of a mathematical statement is another statement that follows easily from it. Sometimes the main theorem of a paper is followed by several corollaries, which advertise the strength of the theorem.

Fermat’s little theorem, which states that if p is a prime and \( a \not \equiv 0 \hspace{2mm} (mod \hspace{1mm} p) \), then \( a^{p-1} \equiv 1 \hspace{2mm} (mod \hspace{1mm} p) \).

Euler’s theorem states that if m is a positive integer and a is another positive integer that is coprime to m(this means that a and m have no common factor), then \( a^{\phi(m)} \equiv 1 \hspace{2mm} (mod \hspace{1mm} m) \). Here \( \phi \) is Euler’s totient function: \( \phi(m) \) is the number of integers less than m that are coprime to m.

Fermat’s little theorem is a corollary of Euler’s theorem. Euler’s theorem is a generalization of Fermat’s little theorem to composite moduli.

Lagrange’s theorem, which states that the size of a group is always divisible by the size of any of its subgroups.

Fermat’s little theorem is also a corollary of Lagrange’s theorem.

Here is the argument in outline. The first step is to show that the numbers 1, 2, . . . , p−1 form a group under multiplication mod p. (This means multiplication followed by taking the remainder on division by p. For example, if p = 7 then the “product” of 3 and 6 is 4, since 4 is the remainder when you divide 18 by 7.) The next step is to note that if \( 1 \leq a \leq p−1 \) then the powers of a (mod p) form a subgroup of this group. Moreover, the size of the subgroup is the smallest positive integer m such that \( a^m \) is congruent to 1 mod p. One then applies Lagrange’s theorem, which states that the size of a group is always divisible by the size of any of its subgroups. In this case, the size of the group is p − 1, from which it follows that p − 1 is divisible by m. But then, since \( a^m = 1 \), it follows that \( a^{p−1} = 1 \).
This argument shows that Fermat’s little theorem is, when viewed appropriately, just one special case of Lagrange’s theorem. (The word “just” is, however, a little misleading, because it is not wholly obvious that the integers mod p form a group in the way stated. This fact is proved using Euclid’s algorithm.) - (The Princeton Companion to Mathematics - Part I Introduction - I.4 The General Goals of Mathematical Research - 3 Generalizing - 3.2 Proving a More Abstract Result)

Definitions

Another very important component of mathematical papers is definitions.

Some mathematicians will tell you that the main aim of their research is to find the right definition, after which their whole area will be illuminated. Yes, they will have to write proofs, but if the definition is the one they are looking for, then these proofs will be fairly straightforward.