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, set 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 integer > 1 can Part 1. About be written as the sum of a prime number 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 integers 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.: Congruent number 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}.