As a service to mathematicians working on proofs or disproofs of major conjectures and problems, we are collecting a set of errors or gaps in arguments that have occured in many attempts. This list is not meant to be exhaustive, but instead to serve as a warning on the challenges of these topics. Additional items to add? Email sjm1@williams.edu.

General comments (see the links below for items related to certain problems)

• Clearly state what your main idea is. Why do you think you are able to succeed when so many others have failed? What is the key insight?

• Some authors introduce new notions and then they use them in their proofs. While the statements of the great conjectures are ofter clear and simple, these intermediate notions can be far less obvious. They must be rigorously and clearly written in compatibility with the usual axioms of Mathematics. A rigorous proof based on not well formed sets, functions etc. cannot be considered as correct. Therefore, a manuscript based on notions which are not well formed within the usual axioms of Mathematics will not be accepted and it is the author's task, not the referee's, to carefully prepare this very important part of the work. Remember you are writing a manuscript for others to read, and your goal is to convince them of the correctness of your approach. You cannot do this until you carefully and rigorously define your terms.

• Make sure when you break into cases that your cases are exhaustive; they do not have to be mutually exclusive but they must cover all possibilities.

• Make sure the conditions of a theorem are met before you use; do not use L'Hopital's rule for example unless you know you have 0/0 or oo/oo.

• Be careful about introducing a lot of notation; authors sometimes forget that certain notations they are using have conditions attached.

• Try to prove special cases. If you are trying to prove Fermat's Last Theorem, do your argument in the special case when n=3 and n=7.

• When doing induction make sure you prove the base case and the inductive step carefully; sometimes authors assume an extra condition in the inductive step. See for example https://www.math.toronto.edu/mathnet/falseProofs/sameAge.html

• Make sure your arguments are rigorous - if you are using words like "clearly", "obviously" and so on that hides a needed step.

• It is fine to motivate arguments from real world systems (such as physics), but that is not a proof.

• These are famous problems - familiarize yourself with the literature, see what others have tried, what partial results they have.

• These problems are among the most important in mathematics, and have been open for decades to millenia. They have stubbornly resisted attack.

• If you are able to prove one of these, try your hand at proving a weaker result and submitting that to another journal for comments.

• When you integrate by parts, make sure you include all boundary terms.

• Be careful interchaning limits, taking limits, .... Respect the limit laws.

• Numerical evidence is not the same as a theoretical proof.

Comments on specific problems.

• Fermat's Last Theorem / Beal's Conjecture

• The Riemann Hypothesis

• The 3x+1 Problem

• Odd Perfect Numbers

• Twin Primes / Goldbach Conjecture