New theorems in undergraduate subjects

[flamengo]

Is it possible to deduce theorems besides those in the books of mathematics (I'm talking about consecrated subjects such as Calculus and Real Analysis, for example)? Again, my question is not at the research level. My question is at the undergraduate level. And if so, why are the Calculus books so standardized ?

 

[mfb]

Depends on what you call theorems. Very general results of interest in many different applications, accessible with undergrad tools: see the textbooks. There are more that you can prove at an undergrad level, but they won't be that important and usually more specialized, otherwise they would have been added to the books.

 

[flamengo]

But I think that sometimes it's necessary to prove a new theorem or lemma in order to solve a problem or exercise. Is this true ? I think this occurs mainly in math competitions.

 

[fresh_42]

But I think that sometimes it's necessary to prove a new theorem or lemma in order to solve a problem or exercise. Is this true ? I think this occurs mainly in math competitions.

These are usually auxiliary statements shaped to fit in the specific situation given and thus not very valuable in general contexts.

There are mainly two types of theorems in textbooks: those which reveal a deeper understanding of a theory and apply to several situations and those needed to prove them, which are often called lemma or proposition. However, there is no norm of how to call what. Some lemmas do actually belong to the first kind and are still called lemma, e.g. for historical reasons.

In the end you could construct any formula that applies to certain numbers and call it theorem. I once found the following funny formula on the internet: $2^n+7^n+8^n+18^n+19^n+24^n=3^n+4^n+12^n+14^n+22^n+23^n \; (n=0,\ldots,5)$
I can't imagine anyone who would call it a theorem or even a proposition. And if it can be found in a textbook, then for the same reason as here: for entertainment.

 

[mfb]

In the end you could construct any formula that applies to certain numbers and call it theorem. I once found the following funny formula on the internet: $2^n+7^n+8^n+18^n+19^n+24^n=3^n+4^n+12^n+14^n+22^n+23^n \; (n=0,\ldots,5)$
I can't imagine anyone who would call it a theorem or even a proposition. And if it can be found in a textbook, then for the same reason as here: for entertainment.

A method of generating similar formulas for even larger n ranges (but with prefactors) is discussed in this thread.