NOTE ON NORMAL DECIMALS
... It was proved by Champernowne [2] that the decimal . 1234567891011 . . . is normal, and by Besicovitch [11 that the same holds for the decimal . 1491625 . . . . Copeland and Erdös [3] have proved that if p i, P2 . . . . is any sequence of positive integers such that, for every 0 < 1, the number of p ...
... It was proved by Champernowne [2] that the decimal . 1234567891011 . . . is normal, and by Besicovitch [11 that the same holds for the decimal . 1491625 . . . . Copeland and Erdös [3] have proved that if p i, P2 . . . . is any sequence of positive integers such that, for every 0 < 1, the number of p ...
Numbers: Fun and Challenge
... • Sum of two squares. Write n = 2f · n1 · n2 , where every prime divisor of n1 (resp. n2 ) is ≡ 1 (mod 4) (resp. ≡ 3 (mod 4)). Fermat showed that r2 (n) > 0 (i.e. n is a sum of two squares) if and only if every prime divisor p of n2 occurs in n2 to an even power. Assume this is the case, Jacobi obta ...
... • Sum of two squares. Write n = 2f · n1 · n2 , where every prime divisor of n1 (resp. n2 ) is ≡ 1 (mod 4) (resp. ≡ 3 (mod 4)). Fermat showed that r2 (n) > 0 (i.e. n is a sum of two squares) if and only if every prime divisor p of n2 occurs in n2 to an even power. Assume this is the case, Jacobi obta ...
Chapter4
... • Even though primes have been studied extensively for centuries, many conjectures about them are unresolved, including: • Goldbach’s Conjecture: Every even integer n, n > 2, is the sum of two primes. It has been verified by computer for all positive even integers up to 1.6 ∙1018. The conjecture is ...
... • Even though primes have been studied extensively for centuries, many conjectures about them are unresolved, including: • Goldbach’s Conjecture: Every even integer n, n > 2, is the sum of two primes. It has been verified by computer for all positive even integers up to 1.6 ∙1018. The conjecture is ...
Lecture 18 1 Pollard`s rho method
... [LJMP90] and Lenstra’s elliptic curve method [Jr.87]. As another application, we will discuss the indexcalculus method for computing discrete logarithms in a later lecture. Smooth numbers also play a crucial role in proving many analytic number theoretic results. To cite an example, the result on in ...
... [LJMP90] and Lenstra’s elliptic curve method [Jr.87]. As another application, we will discuss the indexcalculus method for computing discrete logarithms in a later lecture. Smooth numbers also play a crucial role in proving many analytic number theoretic results. To cite an example, the result on in ...
