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Fermat prime

A prime number of the form $F_k = 2^{2^k}+1$ for a natural number $k$. They are named after Pierre Fermat who observed that $F_0,F_1,F_2,F_3,F_4$ are prime and that this sequence "might be indefinitely extended". To date (2017), no other prime of this form has been found, and it is known, for example, that $F_k$ is composite for $k=5,\ldots,32$. Lucas has given an efficient test for the primality of $F_k$. The Fermat primes are precisely those odd primes $p$ for which a ruler-and-compass construction of the regular $p$-gon is possible: see Geometric constructions and Cyclotomic polynomials.

References

• Richard K. Guy, Unsolved Problems in Number Theory 3rd ed. Springer (2004) ISBN 0-387-20860-7 Zbl 1058.11001
• G.H. Hardy; E.M. Wright. "An Introduction to the Theory of Numbers", Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press (2008) [1938] ISBN 978-0-19-921986-5 Zbl 1159.11001
• Michal Krizek, Florian Luca, Lawrence Somer, "17 Lectures on Fermat Numbers: From Number Theory to Geometry" Springer (2001) ISBN 0-387-21850-5 Zbl 1010.11002

Involution semigroup

A semigroup $(S,{\cdot})$ with an involution $*$, having the properties $(x\cdot y)^* = y^* \cdot x^*$ and $x^{{*}{*}} = x$.

A projection in an involution semigroup is an element $e$ such that $e\cdot e = e = e^*$. There is a partial order on projections given by $e \le f$ if $e\cdot f = e$.

References

• Ivan Rival (ed.),"Algorithms and Order", Kluwer (1989) ISBN 940107691X Zbl 0709.68004

Foulis semigroup

Baer $*$-semigroup

A Baer semigroup with involution.

References

• T.S. Blyth, "Lattices and Ordered Algebraic Structures" Springer (2005) ISBN 1852339055 Zbl 1073.06001
• Ivan Rival (ed.),"Algorithms and Order", Kluwer (1989) ISBN 940107691X Zbl 0709.68004

Isoptic

The locus of intersections of tangents to a given curve meeting at a fixed angle; when the fixed angle is a right angle, the locus is an orthoptic.

The isoptic of a parabola is a hyperbola; the isoptic of an epicycloid is an epitrochoid; the isoptic of a hypocycloid is a hypotrochoid; the isoptic of a sinusoidal spiral is again a sinusoidal spiral; and the isoptic of a cycloid is again a cycloid.

References

• J.D. Lawrence, "A catalog of special plane curves" , Dover (1972) ISBN 0-486-60288-5 Zbl 0257.50002

Nephroid

An epicycloid with parameter $m=2$; an algebraic plane curve with equation $$ x= 3r \cos\theta-r\cos\left[3\theta\right] \,, $$ $$ y= 3r \sin\theta-r\sin\left[3\theta\right] \ . $$

The nephroid is the catacaustic of the cardioid with respect to a cusp, and of a circle with respect to a point at infinity; the evolute of a nephroid is another nephroid.

The nephroid of Freeth is the strophoid of a circle with respect to its centre and a point on the circumference. It has equation $$ r = a(1 + 2\sin(\theta/2)) \ . $$

References

• J.D. Lawrence, "A catalog of special plane curves" , Dover (1972) ISBN 0-486-60288-5 Zbl 0257.50002