A Pythagorean triplet is a set of three positive integers (a, b, c) that satisfies the Pythagorean theorem: a² + b² = c². These integers represent the side lengths of a right-angled triangle. The smallest and most well-known triplet is (3, 4, 5), since 9 + 16 = 25. There are infinitely many Pythagorean triplets.
Pythagorean triplet: three integers (a,b,c) where a²+b²=c².
Smallest triplet: (3,4,5) — 9+16=25.
Formula: for m>1, triplet = (2m, m²−1, m²+1).
Primitive triplets have HCF=1: (3,4,5), (5,12,13), (8,15,17), (7,24,25).
Non-primitive triplets are multiples: (6,8,10), (9,12,15) are multiples of (3,4,5).
Euler's formula generates all primitive triplets: a=m²−n², b=2mn, c=m²+n².
There are infinitely many Pythagorean triplets.
Every even integer ≥ 4 is part of at least one Pythagorean triplet.
A Pythagorean triplet (a, b, c) satisfies: a² + b² = c² where a, b are the two legs and c is the hypotenuse of a right triangle.
Examples: • (3, 4, 5): 3²+4²=9+16=25=5² ✓ • (5, 12, 13): 25+144=169=13² ✓ • (8, 15, 17): 64+225=289=17² ✓ • (7, 24, 25): 49+576=625=25² ✓ • (9, 40, 41): 81+1600=1681=41² ✓ • (6, 8, 10): 36+64=100=10² ✓ (multiple of 3,4,5) • (12, 16, 20): multiple of (3,4,5) • (5, 12, 13), (8, 15, 17), (7, 24, 25) are primitive triplets
For any positive integer m > 1: Triplet: (2m, m² − 1, m² + 1)
Examples using the formula:
m = 2: (4, 3, 5) → i.e., (3, 4, 5) 2m=4; m²−1=3; m²+1=5 4²+3²=16+9=25=5² ✓
m = 3: (6, 8, 10) 2m=6; m²−1=8; m²+1=10 6²+8²=36+64=100=10² ✓
m = 4: (8, 15, 17) 2m=8; m²−1=15; m²+1=17 8²+15²=64+225=289=17² ✓
m = 5: (10, 24, 26) 2m=10; m²−1=24; m²+1=26 10²+24²=100+576=676=26² ✓
Euler's formula (generates ALL primitive triplets): For coprime m > n > 0 with m−n odd: a = m²−n², b = 2mn, c = m²+n²
Primitive triplet: HCF(a, b, c) = 1 — no common factor. Examples: (3,4,5), (5,12,13), (8,15,17), (7,24,25)
Non-primitive triplet: obtained by multiplying a primitive triplet by k. Examples: (6,8,10) = 2×(3,4,5); (9,12,15) = 3×(3,4,5)
First 10 primitive Pythagorean triplets:
Any multiple of a Pythagorean triplet is also a Pythagorean triplet: (3,4,5) × 2 = (6,8,10) (3,4,5) × 3 = (9,12,15) (5,12,13) × 2 = (10,24,26)
A Pythagorean triplet is a set of three positive integers (a, b, c) satisfying a² + b² = c². They represent sides of a right-angled triangle. Example: (3, 4, 5) — 9+16=25=5². Other examples: (5,12,13), (8,15,17), (7,24,25).
For any integer m > 1, the triplet (2m, m²−1, m²+1) is a Pythagorean triplet. Example: m=2 gives (4,3,5); m=4 gives (8,15,17). Euler's general formula: a=m²−n², b=2mn, c=m²+n² generates all primitive triplets.
Yes. 6²+8²=36+64=100=10². It is a non-primitive triplet — it is 2×(3,4,5). Any multiple of a Pythagorean triplet is also a Pythagorean triplet.
Five Pythagorean triplets: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (9,40,41).
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