 This lecture is part of Berkeley Math 115, an introductory undergraduate course on number theory and will be about representing a number N by a binary quadratic form AX squared plus BXY, a CY squared of discriminant D, which is B squared minus four AC. So this will be a continuation of last lecture where we looked at some small discriminants. So last lecture we looked at discriminants minus three, minus four, minus seven, minus eight, minus 11 and minus 12. And what we found was that for each of these discriminants except minus 12, there was a unique form opto equivalence and this made it rather easy to determine which primes are represented by the form. This lecture, we're going to look at the discriminants. D is minus 15, minus 16, minus 19, minus 20 and minus 23. And we will see that for most of these, we start getting extra complications because there's more than one reduced form. So before doing these cases, I'll just quickly recall the results we use. First of all, we know that N is represented primitively by some form of discriminant D if and only if D is a square mod four N. So remember primitive means X, Y, O prime. And the other result we used is that any form is equivalent to a reduced form which satisfies B is less than or A is less than or C in absolute value. And you remember if B was equal to A in absolute value or if A was equal to C, there was some other minor condition about B being at least zero, but we won't worry about that too much until it turns up. So let's first do the case D equals minus 15. So we want to find all reduced forms. So we've got the form AX squared plus BXY plus CY squared with B squared minus four AC is now minus 15. And we've got the reduced condition B is less than or A is less than or C. And you remember these two conditions for positive definite forms implied, three A squared is at most the absolute value of D. And this gives us A must be equal to one or two. And if A is equal to one, well, this condition gives us B odd. So B is equal to plus or minus one. So we get two forms X squared plus XY plus four Y squared and X squared minus XY plus four Y squared. And now you remember that if A and B have the same absolute value, then you can make the change of variable in order to change the sign of B. So we can sort of eliminate this one because A and B have the same absolute value. Now, if we look at reduced forms with A equals two, again, we find B must be equal to plus or minus one. So we get two forms, which are two X squared plus XY plus two Y squared and two X squared minus XY plus two Y squared. And again, we remember that if A is equal to C, then we can change the sign of B and still have the forms properly equivalent. That means we sort of switch X and Y and change the sign of one of them. So we can eliminate this one because A and C have the same absolute value. So we have two forms of discriminant minus 15. And now we see that a prime P is represented by one of these if and only if minus 15 is square modulo four P. And now the cases P equals three and five are kind of special because they divide the discriminant. Primes dividing the discriminant always behave in a slightly funny way. Technically speaking, they turn out to be ramified or whatever. So we're mostly going to ignore the cases P equals three and five. So you can see that if minus 15 is a square modulo four P is not three or five, this just says minus 15 P is equal to plus one. And now we can use the quadratic reciprocity theorem to see that this is equivalent to saying that 15 P, sorry, P 15 is equal to plus one. And this corresponds to P being congruent to one, two, four, or eight modulo 15. By the way, I should have a warning here. This does not imply that P is a quadratic residue of 15 because this is the Jacobi symbol and the thing on the bottom is no longer a prime. And do you remember when the thing on the bottom is no longer a prime, this doesn't necessarily mean P is a quadratic residue. In fact, if P is two mod 15, it's definitely not a quadratic residue of 15. So we see that P is congruent to one, two, four, eight modulo 15 implies P is represented by either x squared plus xy plus four y squared, or two x squared plus xy plus two y squared. And the question is, can we work out which is P represented by? And you can do this by using the following extra trick. What you notice is that this form here is always zero or one modulo three. And this one here is always zero or two modulo three. And so if P is not three or five, then we see that P must be one or two mod three. Now we see if P is one mod three, it can't be represented by this form. So P being congruent to one mod three implies P is of the form x squared plus xy plus four y squared. P is one mod three, and of course it should be one, two, four or eight mod 15. Well, if it's one mod three and one, two, four or eight mod 15, it must be one of these two. So if P is one or four mod 15, it's represented by this. Similarly, if P is congruent to two mod three, which implies since P is one, two, four or eight, it must be P is congruent to two or eight mod 15, then P can be written as two x squared plus xy plus two y squared. So we've now found out which primes are represented by these two forms. We can have a few examples of them. For example, if we take two times three squared plus three times minus one plus minus one squared, we get that's 18, 15. That's two in there, so that's 20, so that's equal to 17. And if we take two times three squared plus three times one plus two times one squared, we get 23. And we notice these are two and eight modulo 15. So this form is indeed representing ones that are two or eight modulo 15. Similarly, we can get some examples. So 19 is four mod 15, so it should be represented by this form here. And indeed we can see that 19 is equal to one squared plus one times two plus four times two squared. And the next one is 31, which I'll just leave for you because I don't see why I should do everything. So in some sense, we can see that one quarter of all primes are represented by this form and one quarter of all primes are represented by this form. Whereas previously, we'd always found that half of all primes were represented by some form. Now let's do D equals minus 16. And this is similar to D equals minus 12. So you remember D equals minus 12 turn out to be more or less the same as D equals minus three with symmetrical complications. And D equals minus 16 will be more or less the same as D equals minus four. Anyway, let's find the reduced forms. So as usual, we have B less than root of A, less than root of C, and B squared minus four AC is minus 16. So now B is even, and we have three A squared is less than root of 16. So A is equal to one or two. And for A equals one, well, B must be even and at most A, so B must be equal to zero. So we get X squared plus four Y squared. So if A is equal to two, then we get B is equal to zero or two. But if B is equal to two, what we're getting is that B squared, so two squared minus four times two times C is equal to minus 16. And this fails because this is divisible by eight. This is divisible by eight and this isn't. So B can't be equal to two. So we get the form two X squared plus two Y squared. So now we get the result. N is represented by either X squared plus four Y squared or two X squared plus two Y squared. That's primitively. If and only if minus 16 is a square modulo N. That's the discriminant modulo four N, sorry. Now let's take N to be a prime and see what happens. Well, this just says minus one is a square modulo P, which is the same as P is congruent to one or two mod four. And the prime P equals two is a special case that's actually represented by this form divides minus 16, so we don't worry about it now. So the other case is a P congruent to one mod four. Now we notice that if prime is congruent to one mod four, it cannot be represented by this form because this form is always even and P is odd. So this says that if P is one mod four, then P is equal to X squared plus four Y squared for some X and Y. So this tells us which primes are of the form X squared plus four Y squared. However, you think about it from this isn't a new result because if P is one mod four, we know it can be written as X squared plus Y squared for some X and Y. And one of these must be even because otherwise, if they were both odd, then this sum would be even. So if say Y is even, we can write this as X squared plus Y over two squared times four. So we already knew that primes the form one mod four of the form X squared plus four Y squared. So that's not a new result. And in general, whenever you multiply some discriminant by some square, then the forms of these two discriminants tend to be related. So, you know, discriminant minus four is equal to minus, is related to discriminant minus four times two squared. And it would also be somewhat related to discriminant minus four times three squared, which is discriminant minus 36 and so on. So now let's do the case D equals minus 20. And as usual, we start off by finding the reduced forms. So we are finding A squared plus B X Y plus C Y squared with B squared minus four AC is minus 20. So we see B is even and we have three A squared is less than or equal to minus D, so plus D. So this means that A is now equal to one or two. And if A is equal to one, as usual, we have B is equal to one. And I'm going to ignore the case B equals minus one because we can get rid of it as usual. Sorry, B is even, so B must be zero, sorry. So we get the form X squared plus five Y squared. And the other case is A is equal to two. And now B must be even and at most A, so we get B is equal to zero or two. And if B is equal to zero, we get zero squared minus four times two times C is equal to minus 20, which is not possible. So B can't be zero. So B is two and we get the form two X squared plus two X Y plus three Y squared. So we see that if minus 20 is a square mod four N, is equivalent to N is primitively represented by one of these. And now as before, let's take N to be a prime. Well, in this case, minus 20 being a square mod four N, and when I say N is a prime, let's take N to be a prime P, which is not equal to two or five because these are the numbers dividing the discriminant which always cause problems. So this now says that minus 20 is a square modulo P, so the Legendre symbol is plus one. And again, by using quadratic reciprocity, this is minus five P is equal to plus one. And now we've got to be a bit careful because this is now three modulo four. So if P is three modulo four, then things go a bit funny. So this is equal to plus one. If P is congruent to one, three, seven or nine modulo 20. So if P is one of these, this means P is represented by either an x squared plus five y squared, or two x squared plus two x y plus three y squared. Now we want to know how can we separate these? Well, what we do is we can look at these modulo primes dividing the discriminant. We can look at these modulo primes dividing the discriminant. And mod two doesn't give us anything terribly useful. But if we look at these mod five, we see that this one here is always one or four mod five unless zero mod five. And this one is always two or three mod five. You can see that by completing the square. So this is congruent to a square mod five, which must be one or four. And by completing the square mod five, we see that this is congruent to two x squared, congruent to two times a square modulo five, so it must be two or three mod five. So we find that if P is congruent to one or four mod five and one of these numbers mod 20, in other words, one or nine mod 20, this implies P is equal to x squared plus five y squared. And P being congruent to three or seven modulo 20, implies P can be written in the form two x squared plus two x y plus three y squared. And we can see some examples of this. For example, if we take P equals 29, this is nine mod 20. So this predicts it should be writable as x squared plus five y squared. And you can see, let me see, if we take three squared plus five times two squared. And on the other hand, if a prime is of the form seven mod 20, then it should be representable in this form. So we can see if we take P equals seven, this can be written as, let me see, two times minus one squared plus, two times minus one times three plus, sorry, I meant to take P equals 23 rather than seven. So two times minus one squared plus two times minus one times three plus three times three squared gives an example of a prime that's three mod 20. Seven is trivial because you can just take x equals y equals one. So we noticed something a little bit funny going on here. When we were looking at forms x squared plus y squared, x squared plus two y squared, x squared plus three y squared and x squared plus seven y squared. And for that matter, x squared plus four y squared, these all represent about half of all primes. However, we notice that x squared plus five y squared represents one quarter of all primes. So we have this sort of, you know, there's no obvious reason why five should behave differently from one, two, three, four, or seven. We haven't done six, but if I remember correctly, six also represents a quarter of all primes. So we get some funny changes in what's going on. Well, now let's look at the case P equals minus 23. So let's find the reduced forms. We have B is less than or equal to A is less than or equal to C and B squared minus four AC is minus 23. And as usual, three A squared is less than or equal to the absolute value of minus 23, which gives us A is equal to one or two. And if A is equal to one, we have B is equal to plus or minus one. And as usual, we ignore the case B equals minus one because of the same absolute value of A. And we find the form x squared plus x y plus six, y squared. If A is equal to two, we find again B is equal to plus or minus one. And now we find two forms, two x squared plus x y plus three y squared and two x squared minus x y plus three y squared. And now you may say that we could eliminate one of these, but we can't really, if we're dealing with proper equivalents, these two forms are actually not properly equivalent. They are improperly equivalent because we can change x to minus x. But we can't, but they're actually the first example of two forms that are improperly equivalent but not properly equivalent. However, they still represent the same numbers. You might think this doesn't matter. So as usual, we see P is represented by one of these if and only if minus 23 is a square modulo for P, which again just says minus 23 P is equal to plus one. And it's easy to see using quadratic reciprocity. These are just the primes of the quadratic residues of 23. And now you may think we could try and separate out these two cases by looking at some sort of congruence, mod 23. But unfortunately, this just doesn't work. You can't separate out these forms by using any congruence. And there doesn't seem to be any neat way of telling which primes are represented by this form and which primes are represented by these two forms. So what's happening is that there are three things that can happen. First of all, there's only one equivalence class. So this is what happened when D was equal to minus three, minus four, minus seven, minus 19 and so on. And this case, it's really easy to tell which primes are represented by the form. Second case we had with things like D equals minus 15 and minus 20, where there were more than one equivalence class, but they can be separated by congruences, modulo sum prime dividing the discriminant. Thirdly, there might be more than one equivalence class and they're hard to separate. So the first example, this is D equals minus 23. And it turns out that this is what almost always happened. These two cases only happen for a finite number of discriminants. They happen for many of the small discriminants, but when the discriminant gets bigger, the third case just takes over. In fact, Gauss actually asked for the problem of finding all cases in which the first case happens. Can you find all discriminants such that there's only one equivalence class of forms? And Gauss found the cases D equals minus three, minus four, minus seven, minus eight, minus 11, minus 19, minus 43, minus 67, minus 163. And he calculated a lot further than that. He did all forms up to several hundred or maybe a thousand, I don't know exactly. He did a massive amount of calculation and found no other forms with only one equivalence class of forms. And Gauss's conjecture was finally solved. Well, the story of who solved it is actually a little bit complicated. So Hayden has sort of had a solution, but it had some possibly had minor gaps and it's not really clear. The first complete proof of Gauss's conjectures by Stark and Baker at about the same time came up with a method that reduced this to a finite amount of calculation. And Stark later very kindly pointed out that Hayden's proof was essentially complete and could easily be fixed. So I just show what happens for the case minus 163. So let's show that if D is equal to minus 163, then all forms are equivalent. So this is a sort of rather more complicated example of the calculation. So we want B squared minus four AC is minus 163 and B less than A is less than C as usual. And this gives us the equality three A squared is less than or equal to 163. And now there are quite a lot of different choices of A. So we get naught less than A is less than or equal to seven. So B is absolute value at most seven. And this means that B squared minus four AC, we're using B squared equals minus four AC is minus 163. So we find AC is equal to 163 plus B squared over four. And now if we take B up to seven, well, we don't have to take all the B's up to seven because we notice from this that B is odd. So B is equal to one, three, five or seven. And now this number here is equal to 41, 43, 47 or 53. And you notice by coincidence, these are all prime. And since AC is a prime number and A is less than or equal to C, this implies that A is equal to one. And this in turn implies B is equal to one. We're ignoring the case minus one. So the only form we get is X squared plus XY plus 41 Y squared. So this is the unique reduced form of discriminant minus 163. So as usual, we could work out which primes are represented by this if we really wanted to. Incidentally, these numbers here are given by the first few values of the expression X squared plus X plus 41, where we're setting Y equal to one. And in fact, this is, it turns out to be a prime for not less than X is less than 40. It's not just a prime for the first four cases, as you're prime for the first 40 cases. And it turns out that whenever D is a discriminant of with only one equivalence class of forms, something similar happens to this that we get X squared plus X plus, this is the discriminant absolute value of the discriminant plus one over four. And this turns out to be prime for a huge number of values. And there are other weird things that happen when there's unique forms of that discriminant. So it's a famous bizarre fact. If you take E to the pi root 163, this is equal to 262537412640768743, who may think this is the world's most boring number, point 9999999999999295. This makes it look a lot more interesting because it's very, very, very, very nearly an integer. And this turns, something like this turns out to happen whenever there's a unique equivalence class of forms with discriminant sum number. So, we showed that 67 was another number. So if you take E to the pi root 67, you're going to get something very close to an integer, although it won't be quite as close as this one here. So, this gets into some fairly deep mathematics involving elliptic curves and something called complex multiplication. So if you want to find out more about this, you can look up the term complex multiplication on Google or Wikipedia or something like that, which will sort of explain why this freaky coincidence is related to class numbers of binary quadratic forms. Okay, that's about all I want to say for the moment about positive definite forms. We haven't yet said very much about indefinite forms, so next lecture I'll be giving some example of indefinite forms such as x squared minus 2y squared and trying to determine which primes can be written in this form.