Appending the Same Digit on the Left Repearedly Appending the Same Digit Repeatedly on the Left of a Positive Integer to Generate a Sequence of Composite Integers.

Background Dr. Lenny Jones tried appending the same digit repeatedly on the right of a positive integer. Abstract. Let k be a positive integer, and suppose that k = a 1 a 2... a t,where a i is the ith digit of k (reading from left to right). Let d ∈ {0, 1,..., 9}.For n ≥ 1, define s n = a 1 a 2... a t ddd…d (this is done n times) Ex. k=67 then the sequence is 67d, 67dd, 67ddd, 67dddd, …..

Background cont. There are many trivial answers. i.e. if d ∈ {0, 1,..., 9} and d is even, then all the terms with be composite. If k=d, then d ∈ {3,7,9} is trivial If d ∈ {5} then the resulting sequence is trivial

Background cont. The only nontrivial cases are when d ∈ {1,3,7,9} AND when the greatest common factor of (k,d)=1 For each d, there are two questions that need to be answered: 1. Does there exist a positive integer k such that s n is composite for all integers n ≥ 1? 2. If the answer to the first question is yes, then can we find the smallest such positive integer k?

Chinese Remainder Theorem Let m 1,m 2,...,m k be positive integers with gcd(m i,m j ) = 1 for all i = j.Let a 1, a 2,..., a k be any integers. Then there exist infinitely many integers x that satisfy all of the congruences x ≡ a i (mod m i ) simultaneously.

Covering System/Covering Def. A (finite) covering system,or simply a covering, of the integers is a system of congruences n ≡ a i (mod m i ), with 1 ≤ i ≤ t, such that every integer n satisfies at least one of the congruences. Ex. 0 mod 2 (covers every even integer) 1 mod 2 (covers every odd integer)

Sierpinski’s Theorem There exist inifinitely many odd positive integers k such that k*2n + 1 is composite for all integers n ≥ 1

The case when d=1 Dr. Jones used the covering: n ≡ 0 (mod 3) n ≡ 2 (mod 3) n ≡ 1 (mod 6) n ≡ 4 (mod 6)

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