• Lemma 76: Let be a convergent to .  Then , and moreover we have .  (The angle bracket notation is the same as that introduced in Lecture 7).  We proved this using our upper bound on , which in turn we proved in Lecture 19.
• Description of the continued-fraction method, which is really a variant of the factor-base method that we saw last time, using Lemma 76 as a crucial ingredient.
• Description of Pollard’s method.

Further reading

Koblitz (A Course in Number Theory and Cryptography) and Davenport (The Higher Arithmetic) both discuss these methods, together with other variants.  Right at the end, I mentioned Shor’s algorithm for factorisation using a quantum computer; Scott Aaronson has written a lovely non-technical description on his blog, and he includes links to a number of sources of more information.

I hope that some of you may be inspired to read in more depth about the ideas of this course, and about other aspects of number theory.  Of course, there are many number theory books and many websites where you can do this, so I shan’t try to list them (but please do share your favourites in the comments below).

I’ll just mention one book, The Princeton Companion to Mathematics, edited by Tim Gowers (with associate editors June Barrow-Green and Imre Leader).  This has short sections on various important concepts in pure mathematics, including several from number theory (such as elliptic curves, the Euclidean algorithm and continued fractions, the ideal class group, L-functions, modular arithmetic, number fields, the Riemann zeta function and many more).  Perhaps even more interestingly, it has slightly longer articles (10–20 pages) on branches of mathematics, written by leading mathematicians in those areas.  So there’s “Algebraic Numbers” by Barry Mazur, “Analytic Number Theory” by Andrew Granville, “Computational Number Theory” by Carl Pomerance, “Arithmetic Geometry” by Jordan S. Ellenberg, and many more (they aren’t all about number theory!).  And then there are short pieces about particular theorems and problems, such as the ABC conjecture, the Birch–Swinnerton-Dyer conjecture, Fermat’s last theorem, the Prime Number Theorem and the Riemann Hypothesis, Problems and Results in Additive Number Theory, and so on.  If you want a broader picture of mathematics and how the ideas from your various lecture courses are used, and if you want to know what current research mathematicians in the area are thinking about, this would be a great place to start reading.  Perhaps your college library has a copy?

Preparation for Lectu… oh, this is the end of the course!

I plan to write about some of the topics from this course in more detail over the next few months (in the style of the `Theorem’ posts on this blog), so you might want to keep an eye out for them.  And of course the comment facility is still there for asking questions.  I hope that this course has inspired you to find out more about Number Theory!

• Description of Fermat factorisation.
• Definition of least absolute residue.  (See also Gauss’s lemma, in lecture 7.)
• Definition of a factor base and of a -number.
• Description of the factor-base method.

Further reading

Koblitz (A Course in Number Theory and Cryptography) and recent editions of Davenport (The Higher Arithmetic) both cover this material nicely.

Preparation for Lecture 24

As we saw, the factor-base method relies on coming up with a suitable factor base and suitable -numbers.  How could continued fractions help us with this?

• Lemma 72: If is not an Euler pseudoprime to some base , then for at least half of all bases we find that is not an Euler pseudoprime.  This was analogous to Proposition 71 last time, and the proof was almost identical.
• Proposition 73: Let be an odd composite natural number.  Then is an Euler pseudoprime to at most half of all bases.  (So there are not numbers analogous to Carmichael numbers.)  We showed that there is some base to which is not an Euler pseudoprime, and then used Lemma 72.  We split into two cases, depending on whether is divisible by the square of a prime or not.
• Description of the Solovay-Strassen primality test, a probabilistic primality test that relies on Proposition 73 for its usefulness.
• Definition of a strong pseudoprime to the base .
• Proposition 74: Let be an odd composite natural number.  If is a strong pseudoprime to the base , then it is also an Euler pseudoprime to the base .  We did not prove this in the lecture.
• Theorem 75: If is an odd composite natural number, then is a strong pseudoprime to the base for at most one quarter of all possible bases.  We did not prove this either.
• Description of the Miller-Rabin primality test, another probabilistic primality test.
• We mentioned the Agrawal-Kayal-Saxena primality test, without going into any details.  See below for a reference.

I gave out the fourth (and final) examples sheet, which is now also available on the course page.  Here are a couple of comments on that examples sheet.

Q5(ii).  You are, of course, supposed to incorporate the condition from part (i) into this statement.  I omitted it to avoid making the question even longer, but this seems to have caused some confusion.  My apologies to those of you who were unclear about this.

Q15.  There is no harm in trying the question as written on the sheet, but I meant to put — the question is more interesting that way!

Further reading

Koblitz (A Course in Number Theory and Cryptography) and Davenport (The Higher Arithmetic) both have material on pseudoprimes and primality testing.  Terry Tao has a blog post about the AKS primality test, with various links to further reading.

Preparation for Lecture 23

Next time we’ll be thinking about factorisation.  Given a large integer , we’d like to find a non-trivial factor.  Why might it help to find two squares that are congruent modulo : that is, and such that ?  And can we always find such squares?

• Lemma 69: If and are solutions to , then so is .  This was a straightforward check (and we managed to avoid doing lots of algebra).  So there are infinitely many solutions to Pell’s equation (since we already know that there is one).
• Definition of a pseudoprime to the base .
• Lemma 70: If is pseudoprime to the bases and , then is pseudoprime to the base and also to the base .  This was easy to check.
• Proposition 71: If is not pseudoprime to some base , then at least half of all bases are such that is not pseudoprime to those bases.  We showed that there is an injection from {bases to which is pseudoprime} to {bases to which is not pseudoprime}.
• Definition of a Carmichael number.
• Definition of an Euler pseudoprime to the base .

Further reading

Baker (A concise introduction to the theory of numbers) and Davenport (The Higher Arithmetic) both discuss Pell’s equation and the link with continued fractions.  Koblitz (A Course in Number Theory and Cryptography) has a good section on primality testing and pseudoprimes.  I mentioned the paper of Alford, Granville and Pomerance that shows that there are infinitely many Carmichael numbers.

Preparation for Lecture 22

We finished with the definition of Euler pseudoprimes.  It would be helpful to look back over our work today on Fermat pseudoprimes, to see whether you can adapt it to Euler pseudoprimes.  I suggest that you start with some examples.  Are there analogues of Proposition 71 and of Carmichael numbers for Euler pseudoprimes?  You could also think about Carmichael numbers: can you find any?  If is a Carmichael number, what can you say about and its prime factors?

• Theorem 63: Let be a natural number, and let have convergents .  If and are integers with , then .  (So .)  The key idea of the proof was to find a way to link and (about which we know very little) with , , and (about which we know quite a lot).  We were able to do this by finding integers and such that and .  By thinking about the signs of and (which must be different), we were able to obtain a lower bound for .
• Corollary 64: If is an integer and is a natural number with , then is a convergent for .  This is one of those proofs that becomes easier if you try to prove something slightly stronger.  We showed that if and , then , and so .
• Definition of Pell’s equation (where is a fixed natural number that is not a square, and we are interested in integer solutions for and ).
• Corollary 65: Let be a natural number that is not a square.  If and are positive integers satisfying Pell’s equation , then is a convergent for .  This follows nicely from the previous result.
• Theorem 66 (Lagrange): The continued fraction of is eventually periodic if and only if is a quadratic irrational.  We did not prove this in the lecture, but saw an outline of some of the ideas of the proof.
• Theorem 67: Let be a natural number that is not a square.  Then has a continued fraction of the form .  We did not prove this either.
• Proposition 68: Let be a natural number that is not a square.  Then there are integers and such that .  We showed that there is a convergent for such that , namely the convergent at the end of either the first or second period.

Further reading

Davenport (The Higher Arithmetic), Baker (A concise introduction to the theory of numbers) and Hardy and Wright (An Introduction to the Theory of Numbers) all cover this material on continued fractions and Pell’s equation.

There’s also some nice material related to this on NRICH, based on Conway’s tangles.  There’s an article by Mike Pearson, which starts by mentioning two NRICH problems; I encourage you to try those problems before reading Mike’s article.

Preparation for Lecture 21

Next time we’ll be thinking about primality testing.  That is, given a large number we wish to check (rapidly!) whether it is prime.  In this course so far we’ve seen a few results of the form “If is prime, then …”, for example Fermat’s Little Theorem and Euler’s criterion.  It would be helpful to go back to those two results in particular, to think about whether the result could hold if is not prime.  Does either result give us a useful primality test?

• Definition of convergents.
• Definition of sequences and .
• Lemma 58: For , we have .  That is, is equal to the convergent.  We proved this by induction, using the fact that .
• Lemma 59: For , we have .  This was an easy induction.
• Lemma 60: For , if , , , …, are integers, then the terms and are coprime integers.  This follows immediately from the previous lemma.
• We reminded ourselves that if we are to find a continued fraction for the irrational number , then we must have , and , and and for .
• Proposition 61: The convergents converge.  That is, if we define and and as above, then the sequence , , , … converges to .  We proved this by using Lemma 58 to show that , and then computing .
• Lemma 62: We have for all , and so we also have for .  We used the expression we obtained for in the previous result, together with a couple of easy estimates using the recurrence for .

Further reading

The usual selection.  Hardy and Wright (An Introduction to the Theory of Numbers) and Baker (A concise introduction to the theory of numbers) both discuss Diophantine approximation via continued fractions, and have some nice material going beyond the topics that we’ll see in lectures.

Preparation for Lecture 20

We’ve seen that continued fractions give a sequence of rational numbers that converge to a given real number (namely the convergents).  Next time we’ll consider these as rational approximations of that real number.  I suggest that you compute some convergents (for some examples that we haven’t seen in lectures), to see whether they give good approximations.  How do they compare with the rational approximations arising from truncating decimal expansions?

Next time we’ll see that in some sense the best rational approximations come from continued fractions.  You could usefully think about how we might formulate that result in a precise way (and, of course, how we might prove it).

We are then going to turn our attention to the Diophantine equation (where is a fixed integer that isn’t a square, and where we are looking for integer solutions and ).  Can you find some solutions when or , for example?  It would be good to get a feeling for whether there are no solutions, some but only finitely many solutions, or infinitely many solutions, for different values of .

(And what happens to the equation when is a square?  Why have I ruled out that case above?)

• We concluded the proof of Bertrand’s postulate from last time.
• Proposition 56 (Dirichlet): Let be a real number and let be a natural number.  Then there is a rational with such that .  We saw that we could prove this quite easily using the pigeonhole principle.
• Definition of continued fractions (finite and infinite), and partial quotients.
• Lemma 57: There is a one-to-one correspondence between finite continued fractions and rational numbers.  We saw the important point that the continued fraction of a rational number must terminate, and saw the link to Euclid’s algorithm.
  

Further reading

Davenport (The Higher Arithmetic), Baker (A concise introduction to the theory of numbers), and Hardy and Wright (An Introduction to the Theory of Numbers) all have good sections on continued fractions, including material beyond what we’ll cover in lectures.

Preparation for Lecture 19

Our next job will be to think about infinite continued fractions.  Can you find a continued fraction for ?  Is it unique?  If you truncate the continued fraction at some point, you get a finite continued fraction and so a rational number.  How does it compare with ?  Investigating these truncated continued fractions before the next lecture would be very helpful.

• Proposition 52 (Legendre‘s formula): For , we have .  We saw that this follows nicely from the inclusion-exclusion principle.
• Lemma 53:  For any natural number , we have .  These are both easy bounds arising from the expansion of or from counting subsets of .
  Lemma 54: For any , we have .  We proved this for natural numbers using induction.
• Theorem 55 (Bertrand‘s postulate): For any natural number , there is a prime with .  Our strategy was to show that the product is strictly bigger than 1.  We noticed that this product divides the binomial coefficient , and found an upper bound for the remainder of the contribution.  Next time we’ll finish studying this upper bound in order to obtain the lower bound on that we wanted.  In the meantime, I encourage you to try to finish the proof for yourself.

Further reading

Erdős and Surányi (Topics in the theory of numbers) discuss Bertrand’s postulate and a number of other interesting results relating to the distribution of the primes.  There’s also a proof of Bertrand’s postulate in Hardy and Wright (An Introduction to the Theory of Numbers).

Preparation for Lecture 18

Of course, you should try to finish the proof of Bertrand’s postulate for yourself.

Then next time we’ll have another change of topic.  We’re going to think about approximating irrational numbers by rationals (Diophantine approximation).  The aim is to get a very good approximation by a rational with very small denominator.  How would you go about finding a good rational approximation to or , for example?  If you find a good approximation, can you be confident that it’s the best option?  (That is, could there be another rational with smaller denominator that gives a better approximation?)

• Theorem 50 (Prime Number Theorem): We have as .  That is, as .  We saw some of the key ideas of the proof, but did not go into detail.
• Definition of the von Mangoldt function .
• Lemma 51: If then .  The left-hand side is the logarithmic derivative of the zeta function.  We saw that we could obtain the right-hand side by taking the logarithm of the Euler product and then differentiating.
• Definition of the Dirichlet series corresponding to a sequence .

I gave out the third examples sheet, which is now available on the course page.  I am grateful to the eagle-eyed student who spotted a typo before the end of the lecture: the first in Q6(i) should of course have been an .  I have corrected this in the online version.  Apologies for any confusion this may have caused.

Further reading

The suggestions I made for the last lecture are just as valid for this lecture.  There’s some nice material on the distribution of the primes in Topics in the Theory of Numbers, by Erdős and Surányi.  There’s some discussion of an elementary proof of the Prime Number Theorem in An Introduction to the Theory of Numbers by Hardy and Wright (at least in the later editions — there was no known elementary proof when the first edition was written!).   Terry Tao has a nice blog post discussing a number of aspects of the distribution of prime numbers.

Preparation for Lecture 17

We’re going to move on to another formula for , the number of primes less than .  How would you compute the number of primes less than 100, or less than 1000 (without a computer, and without just listing all the primes!).  Can you generalise your ideas to find an expression for (an exact expression, not an asymptotic formula)?

• Definition of the Riemann zeta function for .
• Lemma 47: If , then converges absolutely.  Moreover, for any the series converges uniformly for , and so is analytic on .  The important point to remember here is that is complex.  Writing (as is standard, if bizarre), we have .
  
• Proposition 48 (Euler product for ): If , then , where the product is over all primes .  We proved this by considering the ‘partial product’ .  Note that this result essentially records the fundamental theorem of arithmetic.  This Euler product is crucial for linking the zeta function to the primes.
• Lemma 49: If , then .  We proved this using the Euler product, being careful about the infinite product (we showed that for a suitable , we have , and could then check individual factors for primes .
• We noted (without proof) some properties of , such as the continuation to a meromorphic function on the complex plane, and the functional equation.  We saw the Riemann Hypothesis, which asserts that all the non-trivial zeros of lie on the line .
• Definition of the Möbius function .  Exercise: show that is a multiplicative function.
• We noted (without proof) that the Riemann Hypothesis is equivalent to the bound .

Further reading

There are many books that give introductions to the Riemann zeta function.  Davenport’s Multiplicative Number Theory goes into more depth than some.  Ben Green has some online lecture notes from a Part III course a few years ago.  The book Analytic Number Theory by Iwaniec and Kowalski tells you more than you realised you wanted to know about the zeta function and much else besides.

Preparation for Lecture 16

What is the relationship between and ?  Hint: can you write in terms of ?

Can you find an expression for of the form ?  (This latter series is called the Dirichlet series for the sequence .)  The values of will be important when we come to think about the Prime Number Theorem.