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Additional resources for Elementary Number Theory: Some Lecture Notes

Example text

X  n Then we have {b} × S = AZn ∩ ({b} × Zn ) = CZn ∩ ({b} × Zn ) = {b} × C0 · ({b/d} × Zn−1 ). If n = 2, we may use the euclidean algorithm in order to find integers s1 , s2 ∈ Z with s1 a1 + s2 a2 = d, write ai = dci and obtain with x1 x2 = b d s1 s2 +k c2 −c1 ,k ∈ Z all solutions. 15 Sums of two squares and pythagorean triples In this section we discuss our first non-linear diophantine equation: We consider the equation x2 + y 2 = n ∈ N 63 and ask for which n ∈ N there is a solution (x, y) ∈ Z2 .

We note first that CZn = AZn , since column operations correspond to multiplication from the right hand side with elementary matrices and on Zn such matrices act as isomorphisms. Let      x 1    ..  n S :=  .  ∈ Z ; a1 x1 + ... + an xn = b .    x  n Then we have {b} × S = AZn ∩ ({b} × Zn ) = CZn ∩ ({b} × Zn ) = {b} × C0 · ({b/d} × Zn−1 ). If n = 2, we may use the euclidean algorithm in order to find integers s1 , s2 ∈ Z with s1 a1 + s2 a2 = d, write ai = dci and obtain with x1 x2 = b d s1 s2 +k c2 −c1 ,k ∈ Z all solutions.

We note that always |a| ≤ 1 and that for m = apn ∈ Z ⊂ Z(p) with a not divisible with p we have |m| = p−n . 5. Using the absolute we may define the convergence of a sequence of p-adic integers: an → a :⇐⇒ |an − a| → 0. We see that pn → 0 and that any series ∞ an pn n=0 converges. Indeed any p-adic integer has such an expansion with coefficients an = tn ∈ [0, p − 1] ∩ N. Namely if a = ( ν )ν∈N with 0 ≤ ν < pν+1 , then a = lim ν→∞ Now we expand ν ν. t. the basis p and obtain ν ν tn pn , = n=0 where the digits tn satisfy 0 ≤ tn < p and do not depend on ν due to ν+1 ).