Test Your Intuition (16): Euclid’s Number Theory Theorems


Euclid’s book IX on number theory contains 36 propositions.

The 36th proposition is:

Proposition 36.If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect.

It asserts that if 2^n-1 is a prime number then 2^{n-1}\cdot (2^n-1) is a perfect number. (A number m is perfect of it is equal to the sum of its proper divisors.)

This is certainly a remarkable achievement of ancient Greek mathematics. Other Propositions of the same book would be less impressive for us:

Proposition 23.If as many odd numbers as we please are added together, and their multitude is odd, then the sum is also odd.

Proposition 24.If an even number is subtracted from an even number, then the remainder is even.

Proposition 25.If an odd number is subtracted from an even number, then the remainder is odd.

Proposition 26.If an odd number is subtracted from an odd number, then the remainder is even.

Proposition 27.If an even number is subtracted from an odd number, then the remainder is odd.

Proposition 28.If an odd number is multiplied by an even number, then the product is even.

Proposition 29.If an odd number is multiplied by an odd number, then the product is odd.

Test your intuition: What is the reason that deep mathematical results are stated by Euclid along with trivial results.

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16 thoughts on “Test Your Intuition (16): Euclid’s Number Theory Theorems

  1. Andrei Zelevinsky

    Proposition 36 is nice but not deep either. A deeper result (but also not too deep if I remember correctly from my high school years), is that every even perfect number is of this form (Wikipedia attributes it to Euler).

    1. Gil Kalai Post author

      Dear Andrei, in my opinion. the notions of prime numbers and perfect numbers are already quite deep and surprising and so is the (one-side) connection between prime numbers of the form 2^n-1 and even perfect number.

  2. Andrei Zelevinsky

    Dear Gil, well maybe Euclid did not share your opinion, or for him the notions of odd and even numbers were equally deep and surprising? But I am open to another explanation.

  3. Vania Mascioni

    Could it be that algebraic insights were felt as deeper than it is now perceived due to the entirely geometric approach the Greek had to “calculations?” In a way, odd and even do appear as more arcane if all quantities are seen as lengths of segments rather than as elements of the integer number sequence.

  4. Andy D

    With all due respect to the master, maybe Euclid only knew one deep fact about whole numbers. I couldn’t find any others (in my brief perusal of his books—I’m prepared to be wrong).

    Or, since it’s the last entry, maybe Prop. 36 is really a “teaser” for a more advanced volume, that Euclid never (?) got around to publishing.

  5. Joshua Zelinsky

    I’m going to have to agree partially with Vania. Euclid didn’t have good notation for expressing deep result. This is connected to how Euclid also didn’t have a proof of unique prime factorization. He wouldn’t have even had a notation to state it.

    However, there’s another issue at work here that’s also worth pointing out- the other sections of the Elements also have a variety of results some of which are deeper than others, and this applies to the geometric sections of the book as well. Euclid seems to be more interested in building up a large framework and which along the way proves the deep results.

  6. ns

    Some historian said , Euclid’s book is a collection of a lot of researcher’s works.
    Maybe such backgrounds reflects the structure of this book.

  7. Apurv Anand

    I think not all results in the propositions can be attributed to Euclid – it may simply be a matter of putting down things he came across one after the other after the basic ones?
    But there does appear to be some structure in the earlier books which deal with results of planar geometry – propositions do build up on earlier ones – maybe that was all his work and hence better structured.

  8. Kea

    In my opinion the reason is this. In Euclid’s time, there was no doubt a better memory of Pythagorean mathematics, and ancient mathematics in general, than we have even now. But the methodology was new, so Euclid was trying to be thorough, without including his entire body of knowledge, which was lost.

  9. Circe

    This seems to be a hallmark of all ancient mathematics, across cultures, and I think it just shows us how many “non-trivial” things we take for granted today. This guy, for example, who, among other things solved Pell’s equation about half a millennium before the birth of Pell got off to a false start when trying to define arithmetic with 0.

    1. Gil Kalai

      Well, I am not sure the term “the answer” for this historical question is as definite as it is for mathematical questions. Also for the previous “test your intuition” I dont think there is a definite answer. In any case, I hope to resturn with brief comments to both questions sometime soon.

      1. Giovanna

        I searched your blog for a followup and didn’t find one. Perhaps “the” answer is “all necessary truths are ultimately trivial.” That’s sort of what a Tautology is, after all, not merely a necessary truth, but a trivial truth as well. Some may be more elaborate in their explication than others, but they are nevertheless trivial.

  10. Yuri

    As you mention at the beginning of the paragraph, Proposition 36 is the last one from Book IX. so it may be that it is the “final result” of a chain of decuctions, the other propositions being just intermediate rings of this chain. In modern terms, Proposition 36 may be the “theorem”, while the others may be the “claims”.

  11. Johan

    Did not Euclid have a similar goal as Bourbaki, to write a textbook on “all” of mathematics starting from the basics? (Indeed, even the name of Euclid’s book references the title of Borubaki’s work.)

    Presumably Bourbaki has some less deep statement in them as well.


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