Five Unsolved Problems In Number Theory, By Florentin Smarandache

Five Unsolved Problems in Number Theory 1) Let a0Q\{-1,0,1}. Solve for x the equation: x@a1/x + 1x ax = 2a. 2) a) If (a,b) = 1 {are relatively prime numbers}, how many primes does the progression a@ pn+b, for n=1,2,…, contain? Where pn is the nth prime. But how many numbers of the form n! does the progression contain? But of the form nn? b) Same questions for numbers of the form an+b, with aó{"1,0}. c) Same questions for numbers of the form kk+1 and kk-1, with k a natural number. 3) a) Let n be a non-null positive integer and d(n) the number of positive divisors of n. Of course, d(n)#n, and d(n)=1 if and only if n=1. For n≥2 we have d(n)≥2. Find the smallest k, which depends on n, such that: d(d(…d(n)…))=2. --------k times

b) Let F(n) =

G d, and let m be a given positive integer.

d|n, d>0

Find the smallest k, which depends on m, such that F(F(…F(2)…)) ≥ m. ----------k times

4) a) A number is pseudo-prime if some permutation of its digits, including the identity permutation, is a prime. Of course all primes are pseudo-primes. For example 14 is pseudo-prime since a permutation of its digits, 41, is a prime. Now let’s consider the infinite sequence of primes and let’s perform some non-identity permutation of digits for each prime of two or more digits. Do we still get an infinite sequence of primes? b) Similarly, a number is a pseudo-square if some permutation of its digits, including the identity permutation, is a square. Of course all squares are pseudo-squares. For example 52 is pseudo-square since a permutation of its digits, 25, is a square. Now let’s consider the infinite sequence of squares and let’s perform some non-identity permutation of digits for each square of two or more digits. Do we still get an infinite sequence of squares? And similarly for other types of numbers. 5) Let’s construct a sequence using the following binary sieve: - take the set of natural numbers: 1, 2, 3, 4, … ; - delete every second numbers from this set; - delete from the remaining ones every 4th numbers; … and so on: delete, from the remaining ones, every 2k-th numbers, k=1,2,3, … . Is it possible to prove or disprove the following conjectures: a) there are infinitely many primes in this sequence; b) there are infinitely many composite numbers in this sequence. Analogously one can construct a tri-nary sieve, and generally an n-ary sieve. {These problems and conjectures are extracted from the book “Only Problems, Not Solutions!”, by Florentin Smarandache, Chicago, 1991, 1993.} Download more free e-books of number theory from the Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm .