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Zbl 0932.11008
Siler, Joseph R.
Mean crowds and Pythagorean triples.
(English)
[J] Fibonacci Q. 36, No.4, 323-326 (1998). ISSN 0015-0517

The arithmetic, geometric, harmonic and tetra'' means of two positive numbers $p$, $y$ are respectively $a$, $g$, $h$ and $t= (p^2+y^2)/ (p+y)$. The sextuplet $(p, y, t, h, a, g)$ is a mean crowd when its entries are positive integers; it is primitive iff the greatest common divisor of its entries (equivalently $a$ and $h$) is 1. The operator $$(p, y, t, h, a, g)\to (a-g, a+g, a+h, a-h, a, a-p)$$ is an involution on (primitive) mean crowds. It is proved that $(p, y, t, h, a, g)$ is a primitive mean crowd if and only if $p= w^2- wv$ and $y= w^2+ wv$ for some primitive Pythagorean triple $(u, v, w)$.
[Edward J.Barbeau (Toronto)]
MSC 2000:
*11A99 Elementary number theory
11B83 Special sequences of integers and polynomials
11A05 Multiplicative structure of the integers

Keywords: standard binary means; pythagorean triples; mean crowd

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