How can we prove theorems about number theory?

Dec 2018
42
2
Amsterdam
How can we prove theorems about number theory?

Is there an approach for this?
I want to prove multiple theorems
 

romsek

Math Team
Sep 2015
2,969
1,676
USA
what kind of answer are you looking for?
 

romsek

Math Team
Sep 2015
2,969
1,676
USA
I want to formally prove some selected theorems about number theory. How is such a proof constructed and what has to be shown?
a few details might help...

are these theorems that have been proven already?

If nothing else I'd start with a text on number theory and study the proofs in it.
 
Jun 2014
650
54
USA
A statement written in a formal language (of ‘number theory’) has a proof if it is true given the set of axioms (true statements) one chooses to accept as given to be true.

For example, suppose we have an axiom that says 2 is a purple number. Further suppose we have a second axiom that says if $x$ is a purple number, $\exists y = 3x$ such that $y$ is also a purple number. Then we can prove that 18 is a purple number because 6 = 3*2 is a purple number so 18 = 3*6 must also be a purple number.

Generally, in any formal system sufficient to express the natural numbers (arithmetic), there are true statements that cannot be proven within the system. More specifically, see Gödel‘s incompleteness theorem.
 
Aug 2012
2,496
781
A statement written in a formal language (of ‘number theory’) has a proof if it is true given the set of axioms (true statements) one chooses to accept as given to be true.

For example, suppose we have an axiom that says 2 is a purple number. Further suppose we have a second axiom that says if $x$ is a purple number, $\exists y = 3x$ such that $y$ is also a purple number. Then we can prove that 18 is a purple number because 6 = 3*2 is a purple number so 18 = 3*6 mist also be a purple number.

Generally, in any formal system sufficient to express the natural numbers (arithmetic), there are true statements that cannot be proven within the system. More specifically, see Gödel‘s incompleteness theorem.
Your response is technically true but utterly useless. Would you suggest approaching the abc conjecture or the Goldbach conjecture or (before Wiles) FLT? Do you think that if Wiles saw your advice in the late 1980s when he started work on FLT, he would have slapped his forehead and gone, "Duhhhh, why didn't I think of that!"?

Curious as to why you responded as you did, since it's so profoundly different than the way anyone, from freshman to professor, would approach a problem in number theory. It's at the completely wrong level. As if I asked you how to drive a car and you told me I should learn about the chemical engineering of petroleum refineries.
 
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Jun 2014
650
54
USA
Every mathematician offering a proof of a formal statement, regardless of age or experience, must follow the same basic format: that of offering a formal language in which the statement to be proved and its proof can be written along with the axiom(s) necessary to arrive at the proof. Is this not a basic explanation given the OP’s question? Sure, going from universal axioms, such as Peano’s, ZF, those that commonly describe the real numbers, etc., to a proof of something (that had yet to be proven) may involve a bit of work, but that is how all ‘universal’ proofs must be derived. Further explanation may involve noticing that today’s ‘new’ theorems are typically derived from older theorems, which in turn were derived from even older theorems, derived themselves from even older theorems, and so on, so as to ensure that all theorems are ultimately derived from the (basic and relatively universal) set of axioms.
 
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