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Prove that for every positive integer s there exists a positive integer n with the sum of digits in decimal system equal to s which is divisible by s. Prove that there exists an increasing infinite sequence of triangular numbers i. Prove that there exists an increasing infinite sequence of tetrahedral numbers i.
Prove that for every positive integer m every even number 2k can be represented as a difference of two positive integers relatively prime to m. Prove that there exist arbitrarily long arithmetic progressions formed of different positive integers such that every two terms of these progressions are relatively prime.
Prove that for every positive integer k the set of all positive integers n whose number of positive integer divisors is divisible by k contains an infinite arithmetic progression. Find all rectangular triangles with integer sides forming an arithmetic progression. Find an increasing arithmetic progression with the least possible difference, formed of positive integers and containing no triangular number.
Prove by elementary means that each increasing arithmetic progression of positive integers contains an arbitrarily long sequence of consecutive terms which are composite numbers. Prove that for every positive integer s every increasing arithmetic progression of positive integers contains terms with arbitrary first s digits in decimal system.
Find all increasing arithmetic progressions formed of three terms of the Fibonacci sequence see Problem 50 , and prove that there are no increasing arithmetic progressions formed of four terms of this sequence. Find an increasing arithmetic progression with the least difference formed of integers and containing no term of the Fibonacci sequence. Find all arithmetic progressions with difference 10 formed of more than two primes.
Find all arithmetic progressions with difference formed of more than two primes. Find an increasing arithmetic progression with ten terms, formed of primes, with the least possible last term. Give an example of an infinite increasing arithmetic progression formed of positive integers such that no term of this progression can be represented as a sum or a difference of two primes. Find all primes which can be represented both as sums and as differences of two primes.
Find five primes which are sums of two fourth powers of integers. Prove that there exist infinitely many pairs of consecutive primes which are not twin primes. Using the theorem of Lejeune-Dirichlet on arithmetic progressions, prove that there exist infinitely many primes which do not belong to any pair of twin primes.
Find five least positive integers for which n is a product of three different primes. Prove that among each three consecutive integers one has at least two different prime divisors. Prove that there are no four consecutive positive integers with this property. Show by an example that there exist four positive integers such that each of them has exactly two different prime divisors.
For positive integer n, let q,. Using Problem 92, prove that the ratio q,. Prove by elementary means that the Chebyshev theorem implies that for every positive integer s, for all sufficiently large n, between nand 2n there exists at least one number which is a product of s different primes.
Prove that the infinite sequence 1, 31, , ,
250 problems in elementary number theory
250 Problems in Elementary Number Theory - Sierpinski (1970).pdf
250 Problems in Elementary Number Theory, Sierpinski