Calculator Games: Pythagorean Triples

 Pythagorean Triples

The Calculator Fun and Games book probably didn’t cover Pythagorean Triples because back in the mid 80s, pocket calculators didn’t have an x² button.  Also, I didn’t start studying trigonometry and geometry until the late 80s, five or so years after I had the book, so the lack of an x² button on my calculator didn’t even bother me.  The square root button was exciting enough, although it was a little boring because if you hammered it enough it always led to an answer of 1.

However, the concept still holds and it’s possible to study Pythagorean Triples with a basic calculator and a notepad and pen (although a scientific calculator will be helpful).

Pythagoras once decreed that for a right-angled triangle, a² + b² = h².  The square of the hypotenuse, h, is equal to the sum of the squares of the other two sides.  The hypotenuse is the side of the triangle which is opposite the right angle.

Furthermore, we can find an easy solution to a² + b² = h² since 3² + 4² = 5² (9 + 16 = 25).  Integer values of a, b and h (sometimes called c) are called Pythagorean triples.

The challenge is:  can you find any more?

Since Pythagoras did his maths thousands of years ago, many people have studied his work and found plenty more triples.  To list a few:

(3, 4, 5)

3²+4²=9+16=25=5²

(5, 12, 13)

5²+12²=25+144=169=13²

(7, 24, 25)

7²+24²=49+576=625=25²

(8, 15, 17)

8²+15² = 64+225 = 289 =17²

(9, 12, 15)

9²+12²=81+144=225=15²

Others include (20, 21, 29) and (12, 35, 37). 



GENERATING PYTHAGOREAN TRIPLES

It’s possible to generate a set of triples using the formulae:

a = m²−n²

b = 2mn

c = m²+n²

where m and n are positive integers.  For example, m = 4 and n = 2 gives:

a = 12
b = 16
c = 20

Which is a scaled-up version of the 3,4,5 triple (each number is 4* larger than the 3,4,5 triple).  This isn’t a unique triple, it’s just one that’s four times larger than an existing one.

Some observations:

It is not possible to have a Pythagorean triple that contains all prime numbers.  At least one of the numbers must be even (b = 2mn).  m = 2 and n=1 yields 3,4,5.

There are an infinite number of Pythagorean triples (since m and n can have any values).  Many of these are duplicates of other triples, just scaled up (for example, 12,16,20 is a scaled up version of 3,4,5).  

So I’m going to search for ‘unique’ triples that aren’t just multiples of smaller ones, and for this, I suspect that m and n must not have common factors.  For example, m = 4 and n = 2 can be simplified to m = 2 and n = 1 (divide by two), which is how (12,16,20) is related to (3,4,5).  (Wider reading on Pythagorean triples led me to discover that the accepted term is not ‘unique’ but ‘primitive’ triples.  You get the idea.  I have reworded the rest of my article to call them primitive, in line with the rest of the mathematical world).

m = 5, n = 1 gives (10,24,26) or (5,12,13) – which brings me to another observation:  Primitive Pythagorean triples generally contain at least one prime number.  In my list above, I gave (9,12,15) as an example, but that’s just another version of (3,4,5), in this case, scaled up by 3.  By definition,(and I know I haven’t defined them very well) primitive triples must contain three numbers that have no common factors that would enable them to be simplified.  Prime numbers guarantee this, but are not essential – for example, (16, 63, 65) is a triple with no primes.

An easy way to generate primitive triples is to use the m, n method, setting m to an even number and n to 1.  This means that a = m²−n², and c = m²+n² will only have a difference of two.

m	n	a	b	c	a2 + b2	c2
2	1	3	4	5	25	25
4	1	15	8	17	289	289
6	1	35	12	37	1369	1369
8	1	63	16	65	4225	4225
10	1	99	20	101	10201	10201
12	1	143	24	145	21025	21025
14	1	195	28	197	38809	38809
16	1	255	32	257	66049	66049
18	1	323	36	325	105625	105625
20	1	399	40	401	160801	160801
22	1	483	44	485	235225	235225
24	1	575	48	577	332929	332929
26	1	675	52	677	458329	458329
28	1	783	56	785	616225	616225
30	1	899	60	901	811801	811801

Prime numbers highlighted in bold.  Rows of green triples contain no prime numbers.

So: to continue our investigation: what happens if we set n = 2?  Do we still get an abundance of primitive triples?  We do, but only when m is odd.  When m is even and n=2, the values of a,b,c are all divisible by 4 (not shown below).  When m is divisible by 4, then a,b,c are all divisible by 8.

Here are the results for n=2 and m is odd: a series of primitive triples, some containing primes.

m	n	a	b	c	a2 + b2	c2
3	2	5	12	13	169	169
5	2	21	20	29	841	841
7	2	45	28	53	2809	2809
9	2	77	36	85	7225	7225
11	2	117	44	125	15625	15625
13	2	165	52	173	29929	29929
15	2	221	60	229	52441	52441
17	2	285	68	293	85849	85849
19	2	357	76	365	133225	133225
21	2	437	84	445	198025	198025
23	2	525	92	533	284089	284089

Prime numbers highlighted in bold.  Rows of green triples contain no prime numbers.

When n=3, we can generate primitive triples when m is an even number that is not divisible by 3 (m = 4, 8, 10, 14, 16, 20…).  When m and n are both odd, then a,b,c are divisible by 2 (unless m is divisible by 3, when a,b,c are all divisible by 9).  This leads to a complicated repeating pattern, which is easier to show than explain: 

m	n	a	b	c	Notes
4	3	7	24	25	Primitive
5	3	16	30	34	m,n both odd, a,b,c divisible by 2
6	3	27	36	45	multiples of 3, all divisible by 9
7	3	40	42	58	m,n both odd, a,b,c divisible by 2
8	3	55	48	73	Primitive
9	3	72	54	90	m,n, odd and divisible by 3, a,b,c divisible by 18
10	3	91	60	109	Primitive
11	3	112	66	130	m,n both odd, a,b,c divisible by 2
12	3	135	72	153	multiples of 3, all divisible by 9
13	3	160	78	178	m,n both odd, a,b,c divisible by 2
14	3	187	84	205	Primitive
15	3	216	90	234	m,n, odd and divisible by 3, a,b,c divisible by 18
16	3	247	96	265	Primitive
17	3	280	102	298	m,n both odd, a,b,c divisible by 2
18	3	315	108	333	multiples of 3, all divisible by 9
19	3	352	114	370	m,n both odd, a,b,c divisible by 2
20	3	391	120	409	Primitive
21	3	432	126	450	m,n, odd and divisible by 3, a,b,c divisible by 18
22	3	475	132	493	Primitive
23	3	520	138	538	m,n both odd, a,b,c divisible by 2
24	3	567	144	585	multiples of 3, all divisible by 9
25	3	616	150	634	m,n both odd, a,b,c divisible by 2
26	3	667	156	685	Primitive

The investigation (and the pattern-finding) continues, as we try n=4.  Here’s the result – it’s possible to find primitive Pythagorean triples for every case where m = odd again. 

m	n	a	b	c	Notes
5	4	9	40	41	Primitive
6	4	20	48	52	m,n both even: a,b,c divisible by 4
7	4	33	56	65	Primitive
8	4	48	64	80	m,n both divisible by 4, a,b,c divisible by 16
9	4	65	72	97	Primitive
10	4	84	80	116	m,n both even: a,b,c divisible by 4
11	4	105	88	137	Primitive
12	4	128	96	160	m,n both divisible by 4, a,b,c divisible by 32
13	4	153	104	185	Primitive
14	4	180	112	212	m,n both even: a,b,c divisible by 4
15	4	209	120	241	Primitive
16	4	240	128	272	m,n both divisible by 4, a,b,c divisible by 16
17	4	273	136	305	Primitive
18	4	308	144	340	m,n both even: a,b,c divisible by 4
19	4	345	152	377	Primitive
20	4	384	160	416	m,n both divisible by 4, a,b,c divisible by 32

We find that m and n must have no common factors in order to generate primitive triples.  This means m and n must be ‘relatively prime’ (an expression I learned during my wider reading into this project).  

This is starting to go beyond the scope of a calculator, and into spreadsheet territory, but it's still fascinating, and can still be done with a simple pocket calculator (or an app on your phone, naturally), even without an x² button.

Summary:

It is not possible to have a Pythagorean triple that contains all prime numbers

It is possible to have a primitive Pythagorean triple that contains no prime numbers; a, b and c will be relatively prime, and these are generally generated by m and n being relatively prime too.

There are an infinite number of triples, and an infinite number of primitive triples

Previous Calculator Fun and Games articles:

Snakes and Ladders (Collatz Conjecture)
Crafty Calculator Calculations (numerical anagrams, five digits)
More Multiplications (numerical anagrams, four digits)
Over and Out (reduce large numbers to zero in as few steps as possible)
Sums of digits and Multiples of 9