I have another subject nominated by goldenoj today. And it even lets me get into number theory, the field of mathematics questions that everybody understands and nobody can prove.
I was once a young grad student working as a teaching assistant and unaware of the principles of student privacy. Near the end of semesters I would e-mail students their grades. This so they could correct any mistakes and know what they’d have to get on the finals. I was learning Perl, which was an acceptable pastime in the 1990s. So I wrote scripts that would take my spreadsheet of grades and turn it into e-mails that were automatically sent. And then I got all fancy.
It seemed boring to send out completely identical form letters, even if any individual would see it once. Maybe twice if they got me for another class. So I started writing variants of the boilerplate sentences. My goal was that every student would get a mass-produced yet unique e-mail. To best the chances of this I had to make sure of something about all these variant sentences and paragraphs.
So you see the trick. I needed a set of relatively prime numbers. That way, it would be the greatest possible number of students before I had a completely repeated text. We know what prime numbers are. They’re the numbers that, in your field, have exactly two factors. In the counting numbers the primes are numbers like 2, 3, 5, 7 and so on. In the Gaussian integers, these are numbers like 3 and 7 and . But not 2 or 5. We can look to primes among the polynomials. Among polynomials with rational coefficients, is prime. So is . is not.
The idea of relative primes appears wherever primes appears. We can say without contradiction that 4 and 9 are relative primes, among the whole numbers. Though neither’s prime, in the whole numbers, neither has a prime factor in common. This is an obvious way to look at it. We can use that definition for any field that has a concept of primes. There are others, though. We can say two things are relatively prime if there’s a linear combination of them that adds to the identity element. You get a linear combination by multiplying each of the things by a scalar and adding these together. Multiply 4 by -2 and 9 by 1 and add them and look what you get. Or, if the least common multiple of a set of elements is equal to their product, then the elements are relatively prime. Some make sense only for the whole numbers. Imagine the first quadrant of a plane, marked in Cartesian coordinates. Draw the line segment connecting the point at (0, 0) and the point with coordinates (m, n). If that line segment touches no dots between (0, 0) and (m, n), then the whole numbers m and n are relatively prime.
We start looking at relative primes as pairs of things. We can be interested in larger sets of relative primes, though. My little e-mail generator, for example, wouldn’t work so well if any pair of sentence replacements were not relatively prime. So, like, the set of numbers 2, 6, 9 is relatively prime; all three numbers share no prime factors. But neither the pair 2, 6 and the pair 6, 9 are not relatively prime. 2, 9 is, at least there’s that. I forget how many replaceable sentences were in my form e-mails. I’m sure I did the cowardly thing, coming up with a prime number of alternate ways to phrase as many sentences as possible. As an undergraduate I covered the student government for four years’ worth of meetings. I learned a lot of ways to say the same thing.
Which is all right, but are relative primes important? Relative primes turn up all over the place in number theory, and in corners of group theory. There are some thing that are easier to calculate in modulo arithmetic if we have relatively prime numbers to work with. I know when I see modulo arithmetic I expect encryption schemes to follow close behind. Here I admit I’m ignorant whether these imply things which make encryption schemes easier or harder.
Some of the results are neat, certainly. Suppose that the function f is a polynomial. Then, if its first derivative f’ is relatively prime to f, it turns out f has no repeated roots. And vice-versa: if f has no repeated roots, then it and its first derivative are relatively prime. You remember repeated roots. They’re factors like , that foiled your attempt to test a couple points and figure roughly where a polynomial crossed the x-axis.
I mentioned that primeness depends on the field. This is true of relative primeness. Polynomials really show this off. (Here I’m using an example explained in a 2007 Ask Dr Math essay.) Is the polynomial relatively prime to ?
It is, if we are interested in polynomials with integer coefficients. There’s no linear combination of and which gets us to 1. Go ahead and try.
It is not, if we are interested in polynomials with rational coefficients. Multiply by and multiply by . Then add those up.
Tell me what polynomials you want to deal with today and I will tell you which answer is right.
This may all seem cute if, perhaps, petty. A bunch of anonymous theorems dotting the center third of an abstract algebra text will inspire that. The most important relative-primes thing I know of is the abc conjecture, posed in the mid-80s by Joseph Oesterlé and David Masser. Start with three counting numbers, a, b, and c. Require that a + b = c.
There is a product of the unique prime factors of a, b, and c. That is, let’s say a is 36. This is 2 times 2 times 3 times 3. Let’s say b is 5. This is prime. c is 41; it’s prime. Their unique prime factors are 2, 3, 5, and 41; the product of all these is 1,230.
The conjecture deals with this product of unique prime factors for this relatively prime triplet. Almost always, c is going to be smaller than this unique prime factors product. The conjecture says that there will be, for every positive real number , at most finitely many cases where c is larger than this product raised to the power . I do not know why raising this product to this power is so important. I assume it rules out some case where this product raised to the first power would be too easy a condition.
Apart from that bit, though, this is a classic sort of number theory conjecture. Like, it involves some technical terms, but nothing too involved. You could almost explain it at a party and expect to be understood, and to get some people writing down numbers, testing out specific cases. Nobody will go away solving the problem, but they’ll have some good exercise and that’s worthwhile.
And it has consequences. We do not know whether the abc conjecture is true. We do know that if it is true, then a bunch of other things follow. The one that a non-mathematician would appreciate is that Fermat’s Last Theorem would be provable by an alterante route. The abc conjecture would only prove the cases for Fermat’s Last Theorem for powers greater than 5. But that’s all right. We can separately work out the cases for the third, fourth, and fifth powers, and then cover everything else at once. (That we know Fermat’s Last Theorem is true doesn’t let us conclude the abc conjecture is true, unfortunately.)
There are other implications. Some are about problems that seem like fun to play with. If the abc conjecture is true, then for every integer A, there are finitely many values of n for which is a perfect square. Some are of specialist interest: Lang’s conjecture, about elliptic curves, would be true. This is a lower bound for the height of non-torsion rational points. I’d stick to the stuff at a party. A host of conjectures about Diophantine equations — (high school) algebra problems where only integers may be solutions — become theorems. Also coming true: the Fermat-Catalan conjecture. This is a neat problem; it claims that the equation
where a, b, and c are relatively prime, and m, n, and k are positive integers satisfying the constraint
has only finitely many solutions with distinct triplets . The inequality about reciprocals of m, n, and k is needed so we don’t have boring solutions like clogging us up. The bit about distinct triplets is so we don’t clog things up with a or b being 1 and then technically every possible m or n giving us a “different” set. To date we know something like ten solutions, one of them having a equal to 1.
Another implication is Pillai’s Conjecture. This one asks whether every positive integer occurs only finitely many times as the difference between perfect powers. Perfect powers are, like 32 (two to the fifth power) or 81 (three to the fourth power) or such.
So as often happens when we stumble into a number theory thing, the idea of relative primes is easy. And there are deep implications to them. But those in turn give us things that seem like fun arithmetic puzzles.
This closes out the A to Z essays for this week. Tomorrow and Saturday I hope to bring some attention to essays from past years. And next week I figure to open for topics for the end of the alphabet, the promising letters U through Z. This and the rest of the 2019 essays should appear at this link, as should the letter S next Tuesday. And all of the A to Z essays ought to be at this link. Thank you for reading.