Pisot-Vijayaraghavan number
In
mathematics, a
Pisot-Vijayaraghavan number is an
algebraic integer α which is real and exceeds 1, but such that its
conjugate elements are all less than 1 in
absolute value. For example, if α is a
quadratic irrational there is one other conjugate: α′, obtained by changing the sign of the square root in α; from
- α = a + b√d
with
a and
b both integers, or in other cases both half an odd integer, we get
- α′ = a − b√d.
The conditions are then
- α > 1 and - 1< α′ < 1.
This condition is satisfied by the
golden mean φ. We have
- φ = (1 + √5)/2 > 1
and
- φ′ = (1 - √5)/2 = -1/φ.
The general condition was investigated by
G. H. Hardy in relation with a problem of
diophantine approximation. This work was followed up by Tirukkannapuram Vijayaraghavan (
30 November1902 -
20 April1955), an Indian mathematician from the
Madras region who came to Oxford to work with Hardy in the mid-1920s. The same condition also occurs in some problems on
Fourier series, and was later investigated by Pisot. The name now commonly used comes from both of those authors.
See also: Salem number