1. 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.

2. 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, …..

3. 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

4. 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?

5. 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.

6. 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)

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

8. 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)

9. Competition Summarize the competition. Outline your company’s competitive advantage.

10. Goals and Objectives List five-year goals. State specific, measurable objectives for achieving your five- year goals. – List market-share objectives. – List revenue/profitability objectives.

11. Financial Plan Outline a high-level financial plan that defines your financial model and pricing assumptions. – This plan should include expected annual sales and profits for the next three years. – Use several slides to cover this material appropriately.

12. Resource Requirements List requirements for the following resources: – Personnel – Technology – Finances – Distribution – Promotion – Products – Services

13. Risks and Rewards Summarize the risks of the proposed project and how they will be addressed. Estimate expected rewards, particularly if you are seeking funding.

14. Key Issues Near term – Identify key decisions and issues that need immediate or near-term resolution. – State consequences of decision postponement. Long term – Identify issues needing long-term resolution. – State consequences of decision postponement. If you are seeking funding, be specific about any issues that require financial resources for resolution.