Covering systems: number theory in the spirit of Paul Erdos˝ Mark Kozek Covering systems: Part 1. About Paul Erdos˝ number theory in the spirit of Paul Erdos˝ Part 2. Erdos˝ and covering systems in research

Part 3. An Mark Kozek Erdos˝ problem on covering systems Whittier College Part 4: Sierpinski´ and Riesel December 7, 2010 revisited About Paul Erdos.˝

Covering systems: number theory in the spirit of Paul Erdos˝ Mark Kozek

Part 1. About Paul Erdos˝

Part 2. Erdos˝ and covering systems in research

Part 3. An Erdos˝ problem on covering systems

Part 4: Sierpinski´ and Hungarian mathematician, 1913-1996. Riesel revisited Interested in combinatorics, graph theory, number theory, classical analysis, approximation theory, theory, probability theory... More on Paul Erdos.˝

Covering systems: Wrote/co-wrote over 1400 research papers. number theory in the spirit of Paul Erdos˝ Erdos˝ number. Mark Kozek Erdos˝ problems: issued “bounties” for problems that he

Part 1. About thought were interesting or for which he wanted to Paul Erdos˝ know the solution. Part 2. Erdos˝ and covering systems in Erdos˝ resources: research 1 The man who loved only numbers, book by Paul Part 3. An Erdos˝ problem Hoffman, 1998, (based on his Atlantic Monthly article on covering systems from 1989).

Part 4: 2 Sierpinski´ and N is a Number: A portrait of Paul Erdos˝ , film by George Riesel revisited Csicsery, 1993 (also based on the Atlantic Monthly article). 3 And what is your Erdos˝ number?", Caspar Goffman, the American Mathematical Monthly, 1969. Erdos˝ and covering systems in research

Covering systems: number theory in the spirit of Paul Erdos˝ Question: (de Polignac, 1849) Mark Kozek Is it the case that every sufficiently large odd > 1 can

Part 1. About be written as the sum of a and a power of 2? Paul Erdos˝

Part 2. Erdos˝ and covering Some small counter examples include: 127, 905. systems in research (Romanoff) A positive proportion of the may be Part 3. An Erdos˝ problem expressed this way. on covering systems (van der Corput) The exceptions form a set of positive Part 4: density. Sierpinski´ and Riesel revisited (Erdos)˝ Constructed an arithmetic progression of odd integers not representable in this way. Background: (from Number Theory 101)

Covering systems: Def.: theory in the spirit of The integers a and b are congruent modulo the natural Paul Erdos˝ Mark Kozek number n > 1 if there exists an integer, z such that

Part 1. About Paul Erdos˝ a − b = zn.

Part 2. Erdos˝ and covering If so, we write systems in research a ≡ b (mod n). Part 3. An Erdos˝ problem on covering For example, systems

Part 4: 77 ≡ 7 (mod 10) Sierpinski´ and Riesel 500 ≡ 1700 (mod 1200) revisited 3 ≡ 58 (mod 5) 143 ≡ 0 (mod 11). Background: (from Number Theory 101)

Covering systems: number theory in the spirit of Paul Erdos˝ Mark Kozek Def.: Covering system Part 1. About Paul Erdos˝ A covering system or covering, for short, is a finite system of Part 2. Erdos˝ congruences and covering systems in research n ≡ ai (mod mi ), 1 < i ≤ t, Part 3. An Erdos˝ problem on covering systems such that every integer satisfies at least one of the

Part 4: congruences. Sierpinski´ and Riesel revisited Example of a covering.

Covering systems: number theory in the spirit of Paul Erdos˝ Mark Kozek For example, the congruences

Part 1. About Paul Erdos˝ n ≡ 0 (mod 2) Part 2. Erdos˝ ≡ ( ) and covering n 1 mod 3 systems in research n ≡ 3 (mod 4) Part 3. An n ≡ 5 (mod 6) Erdos˝ problem on covering systems n ≡ 9 (mod 12)

Part 4: Sierpinski´ and Riesel form a covering of the integers. revisited More background.

Covering systems: Def.: Chinese Remainder Theorem number theory in the spirit of Paul Erdos˝ A system of congruences has a unique solution if the moduli Mark Kozek are pairwise relatively prime.

Part 1. About Paul Erdos˝ For example: Can we solve for x in the following system?

Part 2. Erdos˝ and covering x ≡ 3 mod 9 systems in research x ≡ 5 mod 10 Part 3. An x ≡ 2 mod 11 Erdos˝ problem on covering systems

Part 4: Sierpinski´ and Yes, because {9,10,11} are pair-wise, relatively prime. We Riesel revisited get: x ≡ 255 mod 990

(Note: 990 = 9 × 10 × 11) Back to de Polignac’s question.

Covering systems: Question: (de Polignac, 1849) number theory in the spirit of > Paul Erdos˝ Is it the case that every sufficiently large odd integer 1 can Mark Kozek be written as the sum of a prime number and a power of 2?

Part 1. About Paul Erdos˝ de Polignac asked about odd numbers of the form Part 2. Erdos˝ n and covering x = p + 2 . systems in research Erdos˝ instead thought about writing primes in the form Part 3. An Erdos˝ problem n on covering p = x − 2 , systems Part 4: and changed the question to: Sierpinski´ and Riesel revisited Question0: (Erdos)˝ Is it possible to find an integer x such that x − 2n is not prime for all (non-negative) integers n? Erdos’˝ approach to de Polignac’s conjecture

Covering systems: This led to the “sub-question”: For which n and for which x number theory n in the spirit of would x − 2 be divisible by 3? Paul Erdos˝ Mark Kozek ⇔ x − 2n = 3z ⇔ x − 2n ≡ 0 (mod 3) ⇔ x ≡ 2n (mod 3)

Part 1. About Paul Erdos˝ Let us take powers of 2 (mod 3) Part 2. Erdos˝ and covering 0 systems in 2 ≡ 1 (mod 3) research 21 ≡ 2 (mod 3) Part 3. An 2 Erdos˝ problem 2 ≡ 4 ≡ 1 (mod 3) on covering 3 systems 2 ≡ 8 ≡ 2 (mod 3)

Part 4: Sierpinski´ and We see that: Riesel revisited 2n 6≡ 0 (mod 3). For even powers of 2, 2n ≡ 1 (mod 3). For odd powers of 2, 2n ≡ 2 (mod 3). Erdos’˝ approach to de Polignac’s conjecture

Covering systems: number theory in the spirit of Paul Erdos˝ This led to the “sub-question”: For which n and for which x Mark Kozek would x − 2n be divisible by 3?

Part 1. About Paul Erdos˝ We have that if, Part 2. Erdos˝ and covering systems in n research n ≡ 0 (mod 2) (even) and x ≡ 1 (mod 3) then x − 2 is Part 3. An divisible by 3. Erdos˝ problem on covering systems OR Part 4: n Sierpinski´ and n ≡ 1 (mod 2) (odd) and x ≡ 2 (mod 3) then x − 2 is Riesel revisited divisible by 3. Erdos’˝ approach to de Polignac’s conjecture

Covering systems: number theory in the spirit of Erdos’˝ strategy was to continue along these lines and try to Paul Erdos˝ find conditions on n and on x that would ensure that x − 2n Mark Kozek would be divisible by primes from a given set (that would Part 1. About include 3). Paul Erdos˝ Part 2. Erdos˝ He found the following relations: and covering systems in n research n ≡ 0 (mod 2) and x ≡ 1 (mod 3) ⇒ 3|x − 2

Part 3. An n Erdos˝ problem n ≡ 0 (mod 3) and x ≡ 1 (mod 7) ⇒ 7|x − 2 on covering n systems n ≡ 1 (mod 4) and x ≡ 2 (mod 5) ⇒ 5|x − 2

Part 4: n Sierpinski´ and n ≡ 3 (mod 8) and x ≡ 8 (mod 17) ⇒ 17|x − 2 Riesel n revisited n ≡ 7 (mod 12) and x ≡ 11 (mod 13) ⇒ 13|x − 2 n ≡ 23 (mod 4) and x ≡ 121 (mod 241) ⇒ 241|x − 2n Erdos’˝ approach to de Polignac’s conjecture

Covering systems: number theory n in the spirit of n ≡ 0 (mod 2) and x ≡ 1 (mod 3) ⇒ 3|x − 2 Paul Erdos˝ n Mark Kozek n ≡ 0 (mod 3) and x ≡ 1 (mod 7) ⇒ 7|x − 2 n Part 1. About n ≡ 1 (mod 4) and x ≡ 2 (mod 5) ⇒ 5|x − 2 Paul Erdos˝ n ≡ 3 (mod 8) and x ≡ 8 (mod 17) ⇒ 17|x − 2n Part 2. Erdos˝ and covering n systems in n ≡ 7 (mod 12) and x ≡ 11 (mod 13) ⇒ 13|x − 2 research n ≡ 23 (mod 24) and x ≡ 121 Part 3. An n Erdos˝ problem (mod 241) ⇒ 241|x − 2 on covering systems We observe that the set of congruences describing n form a Part 4: Sierpinski´ and covering system. Riesel revisited We observe that the set of congruences describing x can be combined solved using Chinese Remainder Theorem. Erdos’˝ arithmetic progression of counterexamples

Covering systems: number theory in the spirit of Paul Erdos˝ By the Chinese Remainder Theorem, we get Mark Kozek

Part 1. About x ≡ 7629217 (mod 11184810), Paul Erdos˝

Part 2. Erdos˝ and covering where systems in research 11184810 = 2 · 3 · 5 · 7 · 13 · 17 · 241.

Part 3. An Erdos˝ problem This gives us the Erdos’˝ arithmetic progression of on covering systems counterexamples to de Polignac’s conjecture.

Part 4: Sierpinski´ and k = 7629217 ± 11184810z, for z ∈ . Riesel Z revisited Erdos’˝ arithmetic progression of counterexamples

Covering systems: number theory in the spirit of There exists an arithmetic progression of odd integers x, Paul Erdos˝ that simultaneously satisfy the system of congruences Mark Kozek

Part 1. About x ≡ 1 (mod 3) Paul Erdos˝ x ≡ 1 (mod 7) Part 2. Erdos˝ and covering x ≡ 2 (mod 5) systems in research x ≡ 8 (mod 17) Part 3. An x ≡ 11 (mod 13) Erdos˝ problem on covering x ≡ 121 (mod 241) systems x ≡ 1 (mod 2), Part 4: Sierpinski´ and n Riesel such that x − 2 is composite (not prime) for all non-negative revisited integers n because x − 2n will be divisible by at least one of the primes from the set {3, 5, 7, 13, 17, 241}.  Question: What next?

Covering systems: number theory in the spirit of Paul Erdos˝ Answer: We try to generalize. Mark Kozek

Part 1. About Recall, Erdos˝ constructed an arithmetic progression of odd Paul Erdos˝ integers x such that x − 2n was composite for all Part 2. Erdos˝ and covering non-negative integers n. systems in research

Part 3. An Erdos˝ problem Def.: Sierpinski´ number on covering systems A Sierpinski´ number is a positive odd integer k with the Part 4: property that k · 2n + 1 is composite for all positive integers Sierpinski´ and Riesel n. revisited Sierpinski´ numbers

Covering systems: Def.: Sierpinski´ number number theory in the spirit of Paul Erdos˝ A Sierpinski´ number is a positive odd integer k with the n Mark Kozek property that k · 2 + 1 is composite for all positive integers n. Part 1. About Paul Erdos˝

Part 2. Erdos˝ and covering systems in research Sierpinski´ (1960) observed the following implications: Part 3. An Erdos˝ problem on covering systems n ≡ 1 (mod 2), k ≡ 1 (mod 3) =⇒ k · 2n + 1 ≡ 0 (mod 3) Part 4: n ≡ 2 (mod 4), k ≡ 1 (mod 5) =⇒ k · 2n + 1 ≡ 0 (mod 5) Sierpinski´ and n ≡ 4 (mod 8), k ≡ 1 (mod 17) =⇒ k · 2n + 1 ≡ 0 (mod 17) Riesel n revisited n ≡ 8 (mod 16), k ≡ 1 (mod 257) =⇒ k · 2 + 1 ≡ 0 (mod 257) n ≡ 16 (mod 32), k ≡ 1 (mod 65537) =⇒ k · 2n + 1 ≡ 0 (mod 65537) n ≡ 32 (mod 64), k ≡ 1 (mod 641) =⇒ k · 2n + 1 ≡ 0 (mod 641) n ≡ 0 (mod 64), k ≡ −1 (mod 6700417) =⇒ k · 2n + 1 ≡ 0 (mod 6700417). Sierpinski´ numbers

Covering systems: number theory in the spirit of The moduli appearing in the congruences involving k are 7 Paul Erdos˝ Mark Kozek primes, the first (perhaps only) five Fermat primes 2n Fn = 2 + 1 for 0 ≤ n ≤ 4 and the two prime of F5. Part 1. About Paul Erdos˝

Part 2. Erdos˝ We add the condition k ≡ 1 (mod 2) to ensure that k is odd. and covering systems in research Then the Chinese Remainder Theorem implies that there Part 3. An are infinitely many Sierpinski´ numbers given by Erdos˝ problem on covering systems k ≡ 15511380746462593381 Part 4: Sierpinski´ and Riesel revisited (mod 2 · 3 · 5 · 17 · 257 · 65537 · 641 · 6700417). Smallest Sierpinski´ number

Covering In 1962, Selfridge (unpublished) found what is believed to systems: number theory be the smallest Sierpinski´ number, namely k = 78557. in the spirit of Paul Erdos˝ Mark Kozek His argument is based on the following implications:

n Part 1. About n ≡ 0 (mod 2), k ≡ 2 (mod 3) =⇒ k · 2 + 1 ≡ 0 (mod 3) Paul Erdos˝ n ≡ 1 (mod 4), k ≡ 2 (mod 5) =⇒ k · 2n + 1 ≡ 0 (mod 5) n ≡ 3 (mod 9), k ≡ 9 (mod 73) =⇒ k · 2n + 1 ≡ 0 (mod 73) Part 2. Erdos˝ n and covering n ≡ 15 (mod 18), k ≡ 11 (mod 19) =⇒ k · 2 + 1 ≡ 0 (mod 19) n systems in n ≡ 27 (mod 36), k ≡ 6 (mod 37) =⇒ k · 2 + 1 ≡ 0 (mod 37) research n ≡ 1 (mod 3), k ≡ 3 (mod 7) =⇒ k · 2n + 1 ≡ 0 (mod 7) n Part 3. An n ≡ 11 (mod 12), k ≡ 11 (mod 13) =⇒ k · 2 + 1 ≡ 0 (mod 13). Erdos˝ problem on covering systems There have been attempts to prove that 78557 is the

Part 4: smallest Sierpinski´ number. Sierpinski´ and Riesel revisited In this regard, the web page http://www.seventeenorbust.com

contains the current up-to-date information. Smallest Sierpinski´ number

Covering As of this writing, there remain 6 values of k < 78557 which systems: number theory are unresolved by the project, namely in the spirit of Paul Erdos˝ Mark Kozek 10223, 21181, 22699, 24737, 55459, 67607.

Part 1. About The most recent value of k < 78557 to have been Paul Erdos˝ eliminated was 33661, by Sturle Sunde’s computer, on Part 2. Erdos˝ and covering October 30, 2007. systems in research

Part 3. An Erdos˝ problem on covering systems

Part 4: Sierpinski´ and Riesel revisited Other generalizations.

Covering systems: number theory in the spirit of Def.: Riesel number Paul Erdos˝ Mark Kozek A Riesel number is a positive odd integer k with the property that k · 2n − 1 is composite for all positive integers Part 1. About Paul Erdos˝ n. Part 2. Erdos˝ and covering systems in The smallest known Riesel number is 509203, due to Riesel research (1956). Part 3. An Erdos˝ problem There have been attempts to prove that 509203 is the on covering systems smallest Riesel number. Part 4: Sierpinski´ and http://www.prothsearch.net/rieselprob.html Riesel revisited As of this writing there remain 64 unresolved candidates, of these 2293 is the smallest. More generalizations

Covering systems: number theory in the spirit of Paul Erdos˝ Conjecture (Chen) Mark Kozek For every positive integer r, there exist infinitely many Part 1. About r n Paul Erdos˝ positive odd integers k such that the number k 2 + 1 has at Part 2. Erdos˝ least two distinct prime factors for each positive integer n. and covering systems in research

Part 3. An Conjecture (Chen) Erdos˝ problem on covering systems For every positive integer r, there exist infinitely many r n Part 4: positive odd integers k such that the number k − 2 has at Sierpinski´ and Riesel least two distinct prime factors for each positive integer n. revisited (Equivalent to k r 2n − 1.) Chen’s conjectures.

Covering Chen (2002) resolves each conjecture in the case that r is systems: number theory odd and in the case that r is twice an odd number and 3 r. in the spirit of - Paul Erdos˝ As he notes, the least r for which his arguments do not Mark Kozek apply are r = 4 and r = 6.

Part 1. About Paul Erdos˝ Conjecture 1 is true in general and that Conjecture 2 holds

Part 2. Erdos˝ in the special cases r = 4 and r = 6. and covering systems in research Theorem (Filaseta, Finch, K., 2008) Part 3. An Erdos˝ problem on covering For every positive integer R, there exist infinitely many systems positive odd numbers k such that each of the numbers Part 4: Sierpinski´ and Riesel n 2 n 3 n R n revisited k2 + 1, k 2 + 1, k 2 + 1,..., k 2 + 1

has at least two distinct prime factors for each positive integer n. The minimum modulus problem

Covering systems: Open problem: (Erdos,˝ $1000) number theory in the spirit of Paul Erdos˝ For every N > 1 does there exist a covering Mark Kozek system with distinct moduli all ≥ N?

Part 1. About Paul Erdos˝

Part 2. Erdos˝ and covering Personal attempts: systems in research Part 3. An 1 Summer 2008, with Kelly Bickell, Michael Firrisa, Juan Erdos˝ problem on covering Pablo Ortiz, and Kristen Pueschel. systems

Part 4: Sierpinski´ and We made it to N = 14. Riesel revisited 2 Summer 2009, with Tobit Raff.

We made it to N = 11. The minimum modulus problem: results

Covering systems: number theory Open problem: (Erdos,˝ $1000) in the spirit of Paul Erdos˝ For every natural number N > 1 does there exist a covering Mark Kozek system with distinct moduli all ≥ N?

Part 1. About Paul Erdos˝ Early results: Part 2. Erdos˝ and covering N = 3: Erdos.˝ systems in research N = 9: Churchhouse, 1968. Part 3. An Erdos˝ problem N = 14: Selfridge. on covering systems N = 18: Krukenburg (Ph.D. Thesis), 1971. Part 4: Sierpinski´ and N = 20: Choi, 1971. Riesel revisited N = 24: Morikawa, 1981. N = 25: Gibson (Ph.D. Thesis) 2008. N = 36, 40: Nielsen, 2009. The minimum modulus problem: techniques

Covering systems: number theory For small N, examples can be worked out by hand, but in the spirit of Paul Erdos˝ quickly computers come into play. Mark Kozek Churchouse’s result (N=9) came from using computers and Part 1. About Paul Erdos˝ a greedy algorithm. The LCM of the moduli was

Part 2. Erdos˝ and covering 7 3 2 systems in 604, 800 = 2 3 5 7. research

Part 3. An Erdos˝ problem on covering systems Krukenburg and Choi’s results did not use computers. The Part 4: LCM of Krukenburg’s moduli was Sierpinski´ and Riesel revisited 475, 371, 719, 222, 400 = 2733527211213217219. Gibson’s techniques

Covering systems: number theory in the spirit of Paul Erdos˝ Gibson uses: Mark Kozek

Part 1. About a greedy algorithm (like Churchhouse) Paul Erdos˝

Part 2. Erdos˝ the notion of an “almost covering” (like Morikawa who in and covering systems in turned used ideas of Krukenburg) research “random covering” (like Erdos)˝ Part 3. An Erdos˝ problem on covering extensive computing. systems

Part 4: Sierpinski´ and Riesel revisited The LCM of the moduli used primes up to 2017. Nielsen’s techniques

Covering systems: number theory Nielsen uses a graph theoretic approach, representing in the spirit of Paul Erdos˝ covering systems as trees, and introducing new primes, as Mark Kozek necessary to “plug” holes.

Part 1. About Paul Erdos˝ The LCM of the moduli uses primes up to 103. Part 2. Erdos˝ and covering systems in Initially, he used the primes in order. research

Part 3. An Erdos˝ problem However, the referee noted that sometimes it was more on covering systems efficient to use certain primes out of order. This allowed for

Part 4: the improvement from N = 36 to N = 40. Sierpinski´ and Riesel revisited There was very little wiggle room, and thus, fears a “negative solution” for the minimum modulus problem. Sierpinski´ + Riesel?

Covering systems: number theory in the spirit of Paul Erdos˝ Mark Kozek Question: Do here exist numbers that are simultaneously Part 1. About Paul Erdos˝ Sierpinski´ and Riesel numbers? Part 2. Erdos˝ and covering systems in Answer: Yes. (Cohen and Selfridge, 1975). research

Part 3. An Erdos˝ problem k = 47867742232066880047611079 on covering systems

Part 4: Sierpinski´ and Riesel revisited Sierpinski´ + Riesel?

Covering systems: number theory in the spirit of However, folks (computer scientists) didn’t read their paper. Paul Erdos˝ So they offered their results. Mark Kozek

Part 1. About Brier (1998) k = 29364695660123543278115025405114452910889 Paul Erdos˝ Gallot (2000) k = 623506356601958507977841221247 Part 2. Erdos˝ and covering Gallot (2000) k = 3872639446526560168555701047 systems in research Gallot (2000) k = 878503122374924101526292469

Part 3. An E. Vantieghem (2010) k = 47867742232066880047611079 Erdos˝ problem on covering Filaseta, Finch and K. (2008) k = 143665583045350793098657 systems

Part 4: Sierpinski´ and Riesel revisited For more information on this problem, visit:

http://www.primepuzzles.net/problems/prob_029.htm Thank you

Covering systems: number theory in the spirit of Paul Erdos˝ Mark Kozek

Part 1. About Paul Erdos˝

Part 2. Erdos˝ and covering Any Questions? systems in research

Part 3. An Erdos˝ problem on covering systems

Part 4: Sierpinski´ and Riesel revisited In Memoriam

Covering systems: number theory in the spirit of Paul Erdos˝ Mark Kozek

Part 1. About Paul Erdos˝

Part 2. Erdos˝ and covering systems in research

Part 3. An Erdos˝ problem on covering systems

Part 4: Sierpinski´ and Riesel revisited John L. Selfridge 1927-2010