MathDB

2

Part of 2001 Bundeswettbewerb Mathematik

Problems(2)

x^2 + p_n * x + q_n is the square of a natural number

Source: Germany Bundeswettbewerb Mathematik 2001, Round 2, Problem 2

1/31/2009
For each n∈N n \in \mathbb{N} we have two numbers pn,qn p_n, q_n with the following property: For exactly n n distinct integer numbers x x the number x^2 \plus{} p_n \cdot x \plus{} q_n is the square of a natural number. (Note the definition of natural numbers includes the zero here.)
number theory unsolvednumber theory
Find explicit function for recursively defined function

Source: Germany Bundeswettbewerb Mathematik 2001, Round 1, Problem 2

2/1/2009
For a sequence ai∈R,i∈{0,1,2,…} a_i \in \mathbb{R}, i \in \{0, 1, 2, \ldots\} we have a_0 \equal{} 1 and a_{n\plus{}1} \equal{} a_n \plus{} \sqrt{a_{n\plus{}1} \plus{} a_n}   \forall n \in \mathbb{N}. Prove that this sequence is unique and find an explicit formula for this recursively defined sequence.
functionquadraticsalgebra unsolvedalgebra