Appending the Same Digit on the Left Repearedly

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 = a1a2 ... at ,where ai is the ith digit of k (reading from left to right). Let d ∈ {0, 1,... , 9}.For n ≥ 1, define • sn= a1a2 ... atddd…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 sn 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 m1,m2,... ,mk be positive integers with gcd(mi ,mj ) = 1 for all i = j.Let a1, a2,... , ak be any integers. Then there exist infinitely many integers x that satisfy all of the congruences x ≡ ai (mod mi ) simultaneously.

6. Covering System/Covering • Def. A (finite) covering system,or simply a covering, of the integers is a system of congruences n ≡ ai (mod mi ), 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.