250 Problems in Elementary Number Theory - Sierpinski (1970)

6 250 PROBLEMS IN NUMBER THEORY65* . . Find an increasing arithmetic progression with the least differenceformed of integers and containing no term of the Fibonacci sequence.66*. Find a progression ak +b (k = 0, 1, 2, ... ), with positive integers aand b such that (a, b) = 1, which does not contain any term of Fibonaccisequence.67. Prove that the arithmetic progression ak+b (k = 0, 1, 2, ... ) withpositive integers a and b such that (a, b) = 1 contains infinitely many termspairwise relatively prime.68*. Prove that in each arithmetic progression ak+b (k = 0, 1, 2, ... )with positive integers a and b there exist infinitely many terms with the sameprime divisors.69. From the theorem of Lejeune-Dirichlet, asserting that each arithmeticprogression ak+b (k = 0, 1, 2, ... ) with relatively prime positive integers aand b contains infinitely many primes, deduce that for every such progressionand every positive integer s there exist infinitely many terms which areproducts of s distinct primes.70. Find all arithmetic progressions with difference 10 formed of morethan two primes.71. Find all arithmetic progressions with difference 100 formed of morethan two primes.72*. Find an increasing arithmetic progression with ten terms, formedof primes, with the least possible last term.73. Give an example of an infinite increasing arithmetic progressionformed of positive integers such that no term of this progression can berepresented as a sum or a difference of two primes.IV. PRIME AND COMPOSITE NUMBERS74. Prove that for every even n > 6 there exist primes p and q suchthat (n-p, n-q) = 1.75. Find all primes which can be represented both as sums and asdifferences of two primes.76. Find three least positive integers n such that there are no primesbetween nand n + 10, and three least positive integers m such that thereare no primes between 10m and lO(m+l).