Fibonacci numbers thus grow very fast with N, indeed in geometric progression. This is often called exponential growth. They remained as curiosities till in the 1960's they were found to be crucial in certain studies in mathematical logic.I suspected they were refering to its use in Hilbert's tenth problem even though that was really 1970 (a quibble) and I would hardly call it crucial (a more substantial objection). In fact Fib Numbers are not even needed in the end. I asked Chris Lastowksi who is a Model Theorist at UMCP and he told me the folowing:
Yes. Matijasec showed that the Fibonacci sequence was diophantine, and this sufficed to solve Hilbert's tenth problem (actually to show it could not be solved), by earlier work of Davis, J. Robinson and Putnam. However, Davis almost immediately showed that the exponential function is diophantine, which yields the solution to H-10 more easily, so I would hardly call that a deep connection.V.S. Varadarajan wanted to make the Fib numbers interesting and important. The attempt was not quite right.
- How bad is it for a history-of-math book to exaggerate how important some concept is?
- How important are the Fib Numbers? Do they come up in Mathematics?
- Could V.S. Varadarajan have picked a better example of their use in mathematics?
- It has been said that the Fib Numbers come up in Nature. According to Fib Flim Flam most of the statements made about Fib numbers and nature are suspect.