Products of random variables and the first digit phenomenon
... where logb a denotes the logarithm in base b of a. The mantissa in base b of a positive real number x is the unique number Mb (x) in [1; b[ such that there exists an integer k satisfying x = Mb (x)bk . When a sequence of positive random variables (Xn ) is of a type usually considered by the probabil ...
... where logb a denotes the logarithm in base b of a. The mantissa in base b of a positive real number x is the unique number Mb (x) in [1; b[ such that there exists an integer k satisfying x = Mb (x)bk . When a sequence of positive random variables (Xn ) is of a type usually considered by the probabil ...
Proof of Infinite Number of Fibonacci Primes Stephen
... By the fundamental theorem of arithmetic, N is divisible by some prime q. Since N is the product of all existing Fibonacci primes plus 1, then this prime q cannot be among the Fi that make up the n Fibonacci primes since by assumption these are all the Fibonacci primes that exist and N is not evenly ...
... By the fundamental theorem of arithmetic, N is divisible by some prime q. Since N is the product of all existing Fibonacci primes plus 1, then this prime q cannot be among the Fi that make up the n Fibonacci primes since by assumption these are all the Fibonacci primes that exist and N is not evenly ...
On integers n for which X n – 1 has divisors of every degree
... that the situation is simpler for squarefree ones. In particular, a squarefree number is ϕ-practical if and only if its canonical prime factorization p1 . . . pk , where p1 < · · · < pk , has each pi ≤ 2 + p1 . . . pi−1 . In this section we shall obtain an asymptotic estimate for the distribution of ...
... that the situation is simpler for squarefree ones. In particular, a squarefree number is ϕ-practical if and only if its canonical prime factorization p1 . . . pk , where p1 < · · · < pk , has each pi ≤ 2 + p1 . . . pi−1 . In this section we shall obtain an asymptotic estimate for the distribution of ...
Applying Gauss elimination from boolean equation systems to
... Example 8. Consider the following boolean equation system: (µx = (y ∧ z) ∨ x) (νy = true) This is not in standard recursive form, as the first equation has “∧” and “∨” symbols in the same equation and a boolean constant still appears in the second equation. We first introduce a new equation for the ...
... Example 8. Consider the following boolean equation system: (µx = (y ∧ z) ∨ x) (νy = true) This is not in standard recursive form, as the first equation has “∧” and “∨” symbols in the same equation and a boolean constant still appears in the second equation. We first introduce a new equation for the ...
Number Theory for Mathematical Contests
... We can say that no history of mankind would ever be complete without a history of Mathematics. For ages numbers have fascinated Man, who has been drawn to them either for their utility at solving practical problems (like those of measuring, counting sheep, etc.) or as a fountain of solace. Number Th ...
... We can say that no history of mankind would ever be complete without a history of Mathematics. For ages numbers have fascinated Man, who has been drawn to them either for their utility at solving practical problems (like those of measuring, counting sheep, etc.) or as a fountain of solace. Number Th ...
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The theretofore unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".