Hensel`s treatment of primitive roots
... That all odd prime powers have primitive roots follows in a beautiful way in Chapter 8 of Hensel’s Zahlentheorie. There are other, more elementary ways to derive the result, but Hensel’s fits wonderfully into the p-adic development of elementary number theory that he gives in this treatise. We can s ...
... That all odd prime powers have primitive roots follows in a beautiful way in Chapter 8 of Hensel’s Zahlentheorie. There are other, more elementary ways to derive the result, but Hensel’s fits wonderfully into the p-adic development of elementary number theory that he gives in this treatise. We can s ...
as a PDF
... For our example we choose 10, so let’s consider 2 · 10; φ(2 · 10) = 2φ(10) = 2(4) = 8. Putting together the two sets mentioned in our previous example we have {1, 3, 7, 9, 11, 13, 17, 19}, exactly all 8 numbers relatively prime to 20. For 80 it holds, but care must be taken; we begin with φ(2 · 40) ...
... For our example we choose 10, so let’s consider 2 · 10; φ(2 · 10) = 2φ(10) = 2(4) = 8. Putting together the two sets mentioned in our previous example we have {1, 3, 7, 9, 11, 13, 17, 19}, exactly all 8 numbers relatively prime to 20. For 80 it holds, but care must be taken; we begin with φ(2 · 40) ...
Solved and unsolved problems in elementary number theory
... Conjecture Only a density zero set of prime numbers appears in the second Euclid–Mullin sequence. ...
... Conjecture Only a density zero set of prime numbers appears in the second Euclid–Mullin sequence. ...
Chapter 4
... • A fraction is in its simplest form (this is also called being expressed in lowest terms) if the Greatest Common Factor (GCF), also called the Greatest Common Divisor (GCD), of the numerator and denominator is 1. For example, 1/2 is in lowest terms but 2/4 is not. ...
... • A fraction is in its simplest form (this is also called being expressed in lowest terms) if the Greatest Common Factor (GCF), also called the Greatest Common Divisor (GCD), of the numerator and denominator is 1. For example, 1/2 is in lowest terms but 2/4 is not. ...
Cryptology
... mod p has exactly two solutions. If p mod 4 =3, then exactly one of these two solutions is also a quadratic residue of p. Theorem [Quadratic Reciprocity] For odd prime numbers p and q, Legendre(p,q)*Legendre(q,p) = (-1)(p-1)(q-1)/4 ...
... mod p has exactly two solutions. If p mod 4 =3, then exactly one of these two solutions is also a quadratic residue of p. Theorem [Quadratic Reciprocity] For odd prime numbers p and q, Legendre(p,q)*Legendre(q,p) = (-1)(p-1)(q-1)/4 ...
Applications of a Continued Fraction Algorithm to Some Class
... and calculate the 77th convergent pn/qn from the quotients an in the standard way. It is shown (e.g. [1, Chapter 33]) that the set {+/?„ ± «í7n}includes all the primitive (i.e., no rational divisors other than ±1) algebraic integers of Q{\/d) with norm less than VCD/4). We adopt from [1] an algorith ...
... and calculate the 77th convergent pn/qn from the quotients an in the standard way. It is shown (e.g. [1, Chapter 33]) that the set {+/?„ ± «í7n}includes all the primitive (i.e., no rational divisors other than ±1) algebraic integers of Q{\/d) with norm less than VCD/4). We adopt from [1] an algorith ...