Numbers

Angle Sums and Pythagorean Triples

I’ve always found it cool that if you double the smaller acute angle in a 3-4-5 triangle you get the larger acute angle in a 7-24-25 right triangle. You can see this as a consequence of the double angle formula for sine. If $\alpha$ is the smaller acute angle in a 3-4-5 triangle, then

$\sin (2\alpha) = 2\sin\alpha\cos\alpha=2\frac{3}{5}\frac{4}{5}=\frac{24}{25}$

In fact, if the sine and cosine of an angle are both rational, then so will be the sine and cosine of twice that angle. This gives a way to turn Pythagorean triples into new Pythagorean triples!

For example, suppose $\alpha$ is an acute angle in a right triangle with $a^2 + b^2 = c^2$ . Then

$\sin 2\alpha = \frac{2ab}{c^2}$
$\cos 2 \alpha = \frac{a^2-b^2}{c^2}$

By the Pythagorean identity

$\left(\frac{2ab}{c^2} \right)^2 + \left(\frac{a^2-b^2}{c^2} \right)^2 = 1$

And so

$\left(2ab \right)^2 + \left(a^2-b^2 \right)^2 = \left(c^2\right)^2$

which of course also follows directly from algebra.

For example, using this process

$(3,4,5) \mapsto (7,24,25) \mapsto(336,527,625)$

Originally posted on Mastodon.

Challenge Numbers

2024 and Differences of Squares — Solution

The new year 2024 is a difference of squares, $2024 = 45^2 - 1^2$ , which got me thinking about a fun little number theory problem:

Is there a largest number that can not be expressed as the difference of squares? If so, find it. If not, prove no such number exists. Good luck, and happy new year!

As promised, here’s my solution.

There are infinitely many numbers that can not be expressed as a difference of squares. In fact, we can completely characterize numbers that can be expressed as a difference of squares and those that can’t. It all starts with factoring.

Differences of squares have a useful structure that can be exposed by factoring:

$a^2 - b^2 = (a + b) \times (a - b)$

We can leverage this structure to answer our question.

Suppose n can be factored as $n = s \times t$ . If n can be expressed as a difference of squares, then we can also write

$n = a^2 - b^2= (a + b) \times (a - b)$

Now set $(a + b) = s$ and $(a - b) = t$ . This gives us the system of equations

$a + b = s$

$a - b = t$

We can solve this system by adding and subtracting the equations. Adding gives us $a = \frac{s+t}{2}$ , and subtracting gives us $b = \frac{s-t}{2}$ . This shows us how to express n as a difference of squares: Just factor n into $s \times t$ , compute $a = \frac{s+t}{2}$ and $b = \frac{s-t}{2}$ , and then $n = (a + b) \times (a - b) = a^2 - b^2$ .

There’s only one thing we have to worry about: a and b must be integers. But as long as s and t have the same parity — that is, s and t are both even or both odd — then the sum and difference of s and t will both be even, and so $\frac{s \pm t}{2}$ will be an integer.

This means that n can be expressed as a difference of squares if and only if we can write $n = s \times t$ where s and t are both odd or both even. This is usually possible, often in multiple ways. But there’s one situation when it isn’t: When n is divisible by 2 exactly once. When this is true, then however you factor n into $s \times t$ , the lone factor of 2 will end up as a part of either s or t, making one of them even and the other odd. Thus, in this case, it’s impossible to factor n so that the two factors have the same parity, and so it’s impossible to express n as a difference of squares.

This gives us a complete answer to our question: A number n is not expressible as a difference of squares if and only if it is divisible by 2 exactly once! In other words, every odd number times 2 is not expressible as a difference of squares, and every other integer is.

As an example, given $n = 105 = 3 \times 5 \times 7$ , we can factor $105 = 7 \times 15$ which gives $a=\frac{15+7}{2}= 11$ and $a=\frac{15-7}{2}=4$ , and sure enough, $105 = 11^2 - 4^2$ . Notice, we could also write $105 = 5 \times 21$ , which gives $a = \frac{21+5}{2}=13$ and $b = \frac{21-5}{2}=8$ , so $105 = 13^2 - 8^2$ .

On the other hand, it isn’t possible to do this at all for 6. There are only two factorizations, $6 = 6 \times 1$ and $6 = 2 \times 3$ , and in both cases the factors have different parity, so the a and b we need won’t be integers.

There’s an interesting resemblance here to Euclid’s Formula for generating Pythagorean triples. There’s also an interesting follow-up question about how many different ways a number can be expressed as a difference of squares. And since the numbers that answer our question are those that are divisible by 2 exactly once, I wonder what properties numbers that are divisible by 3 exactly once have.

Thanks to everyone who contributed on the Mastodon thread! There are some cool ideas there as well, so be sure to check it out.

Challenge Numbers

2024 and Differences of Squares

The new year is one of my favorite kinds of numbers: a difference of squares!

$2024 = 46 \times 44 = (45 + 1) \times (45 - 1) = 45^2 - 1^2$

This observation got me thinking about what kinds of numbers can be written as the difference of squares. For example, $3 = 2^2 - 1^2$ , $5 = 3^2 - 2^2$ , and $16 = 4^2 - 0^2$ , but it is impossible to write 6 as the difference of squares of integers.

So here’s a little mathematical puzzle to start the new year: Is there a largest number that can not be expressed as the difference of squares? If so, find it. If not, prove no such number exists. Good luck, and happy new year! I’ll give my answer in a follow-up post.

Related Posts

2024 and Differences of Squares — Solution
The 2017-2018 Conjecture
Happy 2016!

Numbers Resources Teaching Writing

Pierre de Fermat’s Link to a High School Student’s Prime Math Proof — Quanta Magazine

My latest column for Quanta Magazine tells the mathematical story of the incredible high school student who proved a result about not-quite prime numbers that had eluded mathematicians for decades.

[Daniel] Larsen was a high school student in 2022 when he proved a result about a certain kind of number that had eluded mathematicians for decades. He proved that Carmichael numbers — a curious kind of not-quite-prime number — could be found more frequently than was previously known, establishing a new theorem that will forever be associated with his work. So, what are Carmichael numbers? To answer that, we need to go back in time.

You can read the full article for free here.

By MrHonner, 2 yearsNovember 27, 2023 ago

Numbers Resources Teaching

The Symmetry That Makes Solving Math Equations Easy — Quanta Magazine

My latest column for Quanta Magazine is about one of the most dreaded mathematical objects in high school math: the quadratic formula!

$x=\frac{-b \pm \sqrt{b^2-4ac}} {2a}$

As complicated as the quadratic formula is, the cubic formula is much worse, but a simple geometric idea connects the two.

As intimidating as this looks, hiding inside is a simple secret that makes solving every quadratic equation easy: symmetry. Let’s look at how symmetry makes the quadratic formula work and how a lack of symmetry makes solving cubic equations much, much harder. So much harder, in fact, that a few mathematicians in the 1500s spent their lives embroiled in bitter public feuds competing to do for cubics what was so easily done for quadratics.

You can read the full article here.

By MrHonner, 3 yearsMarch 25, 2023 ago

Angle Sums and Pythagorean Triples

2024 and Differences of Squares — Solution

2024 and Differences of Squares

Pierre de Fermat’s Link to a High School Student’s Prime Math Proof — Quanta Magazine

The Symmetry That Makes Solving Math Equations Easy — Quanta Magazine

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