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Monthly Archives: November 2014
Primitive Heron Triangles and Pythagorean Triples having the same Perimeter
See older post: Heron triangle (a,b,c) – abc(a+b+c) Here’s a list of all Heron triangles with sides less than 100 Pythagorean Triples that have the same perimeter as primitive Heron triangles whose sides are less than 100. … Continue reading
Expressing Consecutive Squares and Cubes
Expressing consecutive squares: …………………………………. …………………………………. Expressing consecutive cubes ……………………………………. ……………………………………. Can you express consecutive 4-th powers?
Integers(x,y,z) such that (xy+x+y),(yz+y+z),(zx+z+x) are all squares
Book III Diophantus Arithmetica: To find integers such that are all squares Solution: , , Indeed,
Primitive Heron triangles (a, b, c) and (s-a, s-b, s-c)
where s is the semiperimeter To find Heron triangle (a, b, c) for which (s-a, s-b, s-c) is also a Heron. N.B. Look for primitive solutions For example, … Continue reading
Generating Primitive Heron triangles from the sides of a PPT
Using these parametric equations to generate primitive Heron triangles I chose all the primitive Pythagorean triples with k ≤ 300
Heron triangle (a,b,c) – abc(a+b+c)
General formula: http://en.wikipedia.org/wiki/Integer_triangle#Heronian_triangles The abovementioned algorithm doesn’t generate, for example, (7, 24, 25). H. Schubert in 1905 published these formulae: which can give us (7, 24, 25). Hermann Schubert’s paper —————————————————————— To find … Continue reading
Pythagorean triple (a,b,c) | Can (c/a + c/b) be an integer?
with equality only if a = b. contradicting a, c both being integers. Thus Now, we need to ask, is there an integer n such that where (a, b, c) is a Pythagorean … Continue reading
Part 3 | Pythagorean triples (8, 15, 17) and (by-ax, ay+bx, cz)
See previous posts: Part 1 | Pythagorean triples (a,b,c), (x,y,z) and (ax-by, ay+bx, cz) Part 2 | Pythagorean triples (5, 12, 13) and (by-ax, ay+bx, cz) Pythagorean triples … Continue reading
Part 2 | Pythagorean triples (5, 12, 13) and (by-ax, ay+bx, cz)
See previous post: Pythagorean triples (a,b,c), (x,y,z) and (ax-by, ay+bx, cz) (a=5, b=12, c=13) and (x, y, z) is any triple from the set of all the primitive Pythagorean triples with c … Continue reading
Pythagorean triples (a,b,c), (x,y,z) and (ax-by, ay+bx, cz)
Let (a, b, c) and (x, y, z) be Pythagorean triples. So, and (by-ax, ay+bx, cz) is a Pythagorean triple (a=3, b=4, c=5) and (x, y, … Continue reading
Fun With Num3ers 