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Pythagorean Triple

Pythagorean Triple

A set of three positive integers which satisfies the Pythagorean theorem a2 + b2 = c2.

 

Examples:
3, 4, 5        32 + 42 = 52
6, 8, 10       62 + 82 = 102
5, 12, 13      52 + 122 = 132
7, 24, 25      72 + 242 = 252
8, 15, 17      82 + 152 = 172
9, 40, 41      92 + 402 = 412
20, 21, 29     202 + 212 = 292

Key Formula

a2+b2=c2a^2 + b^2 = c^2
Where:
  • aa = A positive integer representing one leg of a right triangle
  • bb = A positive integer representing the other leg of a right triangle
  • cc = A positive integer representing the hypotenuse (longest side) of a right triangle

Worked Example

Problem: Verify that (5, 12, 13) is a Pythagorean triple, and then use it to generate a new triple by multiplying each number by 3.
Step 1: Square each of the two smaller numbers and add them together.
52+122=25+144=1695^2 + 12^2 = 25 + 144 = 169
Step 2: Square the largest number and check if it equals the sum.
132=16913^2 = 169
Step 3: Since 169=169169 = 169, the equation a2+b2=c2a^2 + b^2 = c^2 holds, confirming this is a Pythagorean triple.
Step 4: Multiply each value by 3 to generate a new triple.
3×5=15,3×12=36,3×13=393 \times 5 = 15, \quad 3 \times 12 = 36, \quad 3 \times 13 = 39
Step 5: Verify the new triple: 152+362=225+1296=152115^2 + 36^2 = 225 + 1296 = 1521 and 392=152139^2 = 1521. It works.
152+362=39215^2 + 36^2 = 39^2
Answer: (5, 12, 13) is a Pythagorean triple, and multiplying by 3 gives another valid triple: (15, 36, 39).

Another Example

Problem: Use the formula for generating primitive Pythagorean triples to find the triple produced by m = 4 and n = 1.
Step 1: A primitive triple (one where the three numbers share no common factor greater than 1) can be generated using two positive integers m>nm > n with the formulas a=m2−n2a = m^2 - n^2, b=2mnb = 2mn, c=m2+n2c = m^2 + n^2.
Step 2: Substitute m=4m = 4 and n=1n = 1.
a=42−12=16−1=15a = 4^2 - 1^2 = 16 - 1 = 15
Step 3: Calculate bb.
b=2(4)(1)=8b = 2(4)(1) = 8
Step 4: Calculate cc.
c=42+12=16+1=17c = 4^2 + 1^2 = 16 + 1 = 17
Step 5: Verify: 82+152=64+225=289=1728^2 + 15^2 = 64 + 225 = 289 = 17^2. The triple checks out.
82+152=1728^2 + 15^2 = 17^2
Answer: Using m = 4 and n = 1, the generated primitive Pythagorean triple is (8, 15, 17).

Frequently Asked Questions

How many Pythagorean triples are there?
There are infinitely many Pythagorean triples. You can always take any known triple and multiply all three numbers by the same positive integer to get a new one. Beyond that, there are infinitely many primitive triples (where aa, bb, and cc share no common factor), which can be generated using the formula a=m2−n2a = m^2 - n^2, b=2mnb = 2mn, c=m2+n2c = m^2 + n^2 for coprime integers m>nm > n where mm and nn are not both odd.
What is the difference between a Pythagorean triple and a primitive Pythagorean triple?
A primitive Pythagorean triple is one where the three integers share no common factor other than 1. For example, (3,4,5)(3, 4, 5) is primitive because gcd⁡(3,4,5)=1\gcd(3, 4, 5) = 1, but (6,8,10)(6, 8, 10) is not primitive because all three numbers are divisible by 2. Every Pythagorean triple is either primitive or a multiple of a primitive triple.

Why It Matters

Pythagorean triples give you exact integer side lengths for right triangles, which is useful in construction, navigation, and any setting where you need precise measurements without messy decimals. They also appear throughout number theory — Fermat's Last Theorem, one of the most famous results in mathematics, asks whether the equation an+bn=cna^n + b^n = c^n has positive integer solutions for n>2n > 2 (it does not). Knowing common triples like (3,4,5)(3, 4, 5) and (5,12,13)(5, 12, 13) also speeds up standardized test problems involving right triangles.

Common Mistakes

Mistake: Assuming any three numbers that satisfy a2+b2=c2a^2 + b^2 = c^2 form a Pythagorean triple.
Correction: All three values must be positive integers. For instance, (1,1,2)(1, 1, \sqrt{2}) satisfies the equation but is not a Pythagorean triple because 2\sqrt{2} is not an integer.
Mistake: Thinking there is only a small, fixed list of Pythagorean triples.
Correction: There are infinitely many. You can generate new ones by scaling (multiplying all three numbers by the same integer) or by using the formula a=m2−n2a = m^2 - n^2, b=2mnb = 2mn, c=m2+n2c = m^2 + n^2 with different choices of mm and nn.

Related Terms