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