# 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

Posted in Number Puzzles | Tagged | 3 Comments

## 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

Posted in Number Puzzles | Tagged | 2 Comments

## 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