Task: Text Categorization
Input (English): Carmichael proved that large ones of these do not share prime factors with other ones, and Wall-Sun-Sun primes are necessarily these. Matiyasevich's disproof of Hilbert's tenth problem relied on these values to show that Diophantine equations have solutions that can grow exponentially, and a pair of them give the Euclidean algorithm its worst-case runtime. Their generating function is x divided by 1 minus x minus x squared, and every positive integer can be written as the sum of non-consecutive ones of these according to Zeckendorf's theorem. The Lucas numbers obey a similar rule as them, and one formula for generating them relies on the Golden Ratio, known as Binet's formula. Defined recursively by summing the previous two numbers in the sequence, for 10 points, name these numbers that begin 1, 1, 2, 3, 5
Output (English): Science