By James K. Strayer

During this student-friendly textual content, Strayer provides all the themes valuable for a primary path in quantity concept. also, chapters on primitive roots, Diophantine equations, and persevered fractions permit teachers the pliability to tailor the fabric to satisfy their very own lecture room wishes. each one bankruptcy concludes with seven pupil initiatives, considered one of which regularly comprises programming a calculator or computing device. the entire tasks not just have interaction scholars in fixing number-theoretical difficulties but in addition support familiarize them with the appropriate mathematical literature.

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**Additional resources for Elementary Number Theory**

**Example text**

The contraction principle) was used for this purpose by Picard (1890): Proposition 28 Let t0 ∈ R, ξ0 ∈ Rn and let U be a neighbourhood of (t0 , ξ0 ) in R × Rn . If ϕ : U → Rn is a continuous map with a derivative ϕ ′ with respect to x that is continuous in U , then the differential equation d x/dt = ϕ(t, x) (1) has a unique solution x(t) which satisfies the initial condition x(t0 ) = ξ0 and is defined in some interval |t − t0 | ≤ δ, where δ > 0. (2) 4 Metric Spaces 37 Proof If x(t) is a solution of the differential equation (1) which satisfies the initial condition (2), then by integration we get t x(t0 ) = ξ0 + ϕ[τ, x(τ )]dτ.

This is not true, however, for Proposition 25. An ordered field need not have the least upper bound property, even though every fundamental sequence is convergent. It is true, however, that an ordered field has the least upper bound property if and only if it has the Archimedean property (Proposition 19) and every fundamental sequence is convergent. In a course of real analysis one would now define continuity and prove those properties of continuous functions which, in the 18th century, were assumed as ‘geometrically obvious’.

But x 2 (t) = E(τ )E(t) satisfies the same differential equation and the same initial condition. e. E(t + τ ) = E(t)E(τ ). (4) In particular, E(t)E(−t) = 1, E(2t) = E(t)2 . The last relation may be used to extend the definition of E(t), so that it is continuously differentiable and a solution of (3) also for |t| < 2R. It follows that the solution E(t) is defined for all t ∈ R and satisfies the addition theorem (4) for all t, τ ∈ R. It is instructive to carry through the method of successive approximations explicitly in this case.