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