Need books/articles with proofs of polygonal number theorem

In summary, the conversation discusses the need for resources on Fermat's polygonal number theorem, including books that provide an exposition and history of the theorem as well as a proof. The conversation mentions acquiring Nathanson's Additive Number Theory and a recommendation for Stillwell's "Mathematics and Its History" which mentions the theorem and its proofs by Cauchy and Nathanson. The preference is for elementary proofs as the speaker has little knowledge of number theory.
  • #1
Wretchosoft
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I am giving a short presentation on Fermat's polygonal number theorem (any number may be written as the sum of n n-gonal numbers). I need books that provide some exposition/history on the theorem as well as a proof. I acquired Nathanson's Additive Number Theory from my university's library, but I'm not sure where to find more on the subject.

Oh, and the proofs should preferably be elementary, as I really know no number theory at all.
 
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  • #2
Wow, quite a coincidence - I just read chapter 3 of Stillwell, "Mathematics and Its History", which mentions this result! He says it was proved by Cauchy in 1813, with a "short" proof by Nathanson in 1987 (Proc Am Math Soc, 99, 22-24).

good luck!
 

1. What is the polygonal number theorem?

The polygonal number theorem, also known as the Cauchy polygonal number theorem, states that every positive integer can be written as the sum of at most 3 polygonal numbers, where a polygonal number is a number that can be represented as the dots or points in a regular polygon.

2. Why is it important to have proofs of the polygonal number theorem?

Having proofs of the polygonal number theorem is important because it provides a mathematical basis for understanding the relationship between polygonal numbers and positive integers. It also allows us to make conjectures and prove other related theorems.

3. Can you explain the proof of the polygonal number theorem?

The proof of the polygonal number theorem involves using a geometric series to represent a polynomial function, which can then be manipulated to show that every positive integer can be written as the sum of at most 3 polygonal numbers. This involves using algebraic manipulations and number theory concepts.

4. Are there any real-world applications of the polygonal number theorem?

While the polygonal number theorem may not have direct real-world applications, it is a fundamental concept in mathematics and can be applied to various fields such as cryptography, number theory, and geometry.

5. Are there any variations of the polygonal number theorem?

Yes, there are variations of the polygonal number theorem, such as the generalized polygonal number theorem, which states that every positive integer can be written as the sum of at most n polygonal numbers, where n is any positive integer. There are also variations for specific types of polygonal numbers, such as triangular numbers or square numbers.

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