Fibonacci & Primes

Tests whether a number belongs to the Fibonacci sequence, gives its position and nearest neighbors, and factors any number into primes, written out and in exponent form. The test is Ira Gessel’s: n is a Fibonacci number exactly when 5n² + 4 or 5n² − 4 is a perfect square. The sequence is named after Leonardo of Pisa, whose Liber abaci (1202) reaches it through a problem about breeding rabbits.

• J. J. O’Connor and E. F. Robertson, “Leonardo Pisano Fibonacci”, MacTutor History of Mathematics (University of St Andrews)
• Ira Gessel, Problem H-187, The Fibonacci Quarterly 9.5 (1971); solution “Fibonacci is a Square”, 10.4 (1972): 417–419