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On the periodicity of irreducible elements in arithmetical congruence monoids Christopher ONeill University of California Davis coneill@math.ucdavis.edu Joint with Jacob Hartzer (undergraduate) Jan 6, 2017 Christopher ONeill (UC Davis)

  1. On the periodicity of irreducible elements in arithmetical congruence monoids Christopher O’Neill University of California Davis coneill@math.ucdavis.edu Joint with Jacob Hartzer (undergraduate) Jan 6, 2017 Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 1 / 8

  2. Arithmetical congruence monoids (ACMs) Definition An arithmetical congruence monoid is a multiplicative set M a , b = { a , a + b , a + 2 b , a + 3 b , . . . } ⊂ ( Z ≥ 1 , · ) for 0 < a < b with a 2 ≡ a mod b . Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 2 / 8

  3. Arithmetical congruence monoids (ACMs) Definition An arithmetical congruence monoid is a multiplicative set M a , b = { a , a + b , a + 2 b , a + 3 b , . . . } ⊂ ( Z ≥ 1 , · ) for 0 < a < b with a 2 ≡ a mod b . Example The Hilbert monoid M 1 , 4 = { 1 , 5 , 9 , 13 , 17 , 21 , 25 , 29 , 33 , . . . } . Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 2 / 8

  4. Arithmetical congruence monoids (ACMs) Definition An arithmetical congruence monoid is a multiplicative set M a , b = { a , a + b , a + 2 b , a + 3 b , . . . } ⊂ ( Z ≥ 1 , · ) for 0 < a < b with a 2 ≡ a mod b . Example The Hilbert monoid M 1 , 4 = { 1 , 5 , 9 , 13 , 17 , 21 , 25 , 29 , 33 , . . . } . 65 = 5 · 13 Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 2 / 8

  5. Arithmetical congruence monoids (ACMs) Definition An arithmetical congruence monoid is a multiplicative set M a , b = { a , a + b , a + 2 b , a + 3 b , . . . } ⊂ ( Z ≥ 1 , · ) for 0 < a < b with a 2 ≡ a mod b . Example The Hilbert monoid M 1 , 4 = { 1 , 5 , 9 , 13 , 17 , 21 , 25 , 29 , 33 , . . . } . 65 = 5 · 13 (prime in Z ⇒ irreducible in M 1 , 4 ). Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 2 / 8

  6. Arithmetical congruence monoids (ACMs) Definition An arithmetical congruence monoid is a multiplicative set M a , b = { a , a + b , a + 2 b , a + 3 b , . . . } ⊂ ( Z ≥ 1 , · ) for 0 < a < b with a 2 ≡ a mod b . Example The Hilbert monoid M 1 , 4 = { 1 , 5 , 9 , 13 , 17 , 21 , 25 , 29 , 33 , . . . } . 65 = 5 · 13 (prime in Z ⇒ irreducible in M 1 , 4 ). 9 , 21 , 49 ∈ M 1 , 4 are irreducible. Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 2 / 8

  7. Arithmetical congruence monoids (ACMs) Definition An arithmetical congruence monoid is a multiplicative set M a , b = { a , a + b , a + 2 b , a + 3 b , . . . } ⊂ ( Z ≥ 1 , · ) for 0 < a < b with a 2 ≡ a mod b . Example The Hilbert monoid M 1 , 4 = { 1 , 5 , 9 , 13 , 17 , 21 , 25 , 29 , 33 , . . . } . 65 = 5 · 13 (prime in Z ⇒ irreducible in M 1 , 4 ). 9 , 21 , 49 ∈ M 1 , 4 are irreducible. 441 = 9 · 49 = 21 · 21 Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 2 / 8

  8. Arithmetical congruence monoids (ACMs) Definition An arithmetical congruence monoid is a multiplicative set M a , b = { a , a + b , a + 2 b , a + 3 b , . . . } ⊂ ( Z ≥ 1 , · ) for 0 < a < b with a 2 ≡ a mod b . Example The Hilbert monoid M 1 , 4 = { 1 , 5 , 9 , 13 , 17 , 21 , 25 , 29 , 33 , . . . } . 65 = 5 · 13 (prime in Z ⇒ irreducible in M 1 , 4 ). 9 , 21 , 49 ∈ M 1 , 4 are irreducible. 441 = 9 · 49 = 21 · 21 = (3 2 ) · (7 2 ) = (3 · 7) · (3 · 7). Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 2 / 8

  9. ACM software package ArithmeticalCongruenceMonoid : a Sage package, available from https://www.math.ucdavis.edu/~coneill/acms/ Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 3 / 8

  10. ACM software package ArithmeticalCongruenceMonoid : a Sage package, available from https://www.math.ucdavis.edu/~coneill/acms/ sage: load( ' /.../ArithmeticalCongruenceMonoid.sage ' ) sage: H = ArithmeticalCongruenceMonoid(1, 4) sage: H Arithmetical Congruence Monoid (1, 4) Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 3 / 8

  11. ACM software package ArithmeticalCongruenceMonoid : a Sage package, available from https://www.math.ucdavis.edu/~coneill/acms/ sage: load( ' /.../ArithmeticalCongruenceMonoid.sage ' ) sage: H = ArithmeticalCongruenceMonoid(1, 4) sage: H Arithmetical Congruence Monoid (1, 4) sage: H.Factorizations(47224750041) [[17, 21, 49, 89, 30333], [17, 21, 21, 89, 70777], [9, 17, 49, 89, 70777]] Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 3 / 8

  12. ACM software package ArithmeticalCongruenceMonoid : a Sage package, available from https://www.math.ucdavis.edu/~coneill/acms/ sage: load( ' /.../ArithmeticalCongruenceMonoid.sage ' ) sage: H = ArithmeticalCongruenceMonoid(1, 4) sage: H Arithmetical Congruence Monoid (1, 4) sage: H.Factorizations(47224750041) [[17, 21, 49, 89, 30333], [17, 21, 21, 89, 70777], [9, 17, 49, 89, 70777]] sage: H.IsIrreducible(999997) # takes a few seconds False Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 3 / 8

  13. ACM software package ArithmeticalCongruenceMonoid : a Sage package, available from https://www.math.ucdavis.edu/~coneill/acms/ sage: load( ' /.../ArithmeticalCongruenceMonoid.sage ' ) sage: H = ArithmeticalCongruenceMonoid(1, 4) sage: H Arithmetical Congruence Monoid (1, 4) sage: H.Factorizations(47224750041) [[17, 21, 49, 89, 30333], [17, 21, 21, 89, 70777], [9, 17, 49, 89, 70777]] sage: H.IsIrreducible(999997) # takes a few seconds False sage: H.IrreduciblesUpToElement(10000001) sage: H.IsIrreducible(999997) # immediate False Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 3 / 8

  14. 500 1000 1500 2000 2500 reducible irreducible Periodicity in ACMs Question [Baginski–Chapman, 2014] When is the list of irreducibles in M a , b (eventually) periodic? Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 4 / 8

  15. 500 1000 1500 2000 2500 reducible irreducible Periodicity in ACMs Question [Baginski–Chapman, 2014] When is the list of irreducibles in M a , b (eventually) periodic? Use IrreduciblesUpToElement() to precompute reducible elements: Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 4 / 8

  16. 1000 500 1500 2000 2500 reducible irreducible Periodicity in ACMs Question [Baginski–Chapman, 2014] When is the list of irreducibles in M a , b (eventually) periodic? Use IrreduciblesUpToElement() to precompute reducible elements: M 1 , 4 : 1 , 25 , 45 , 65 , 81 , 85 , . . . M 5 , 20 : 25 , 125 , 225 , 325 , 425 , 525 , . . . M 7 , 42 : 49 , 343 , 637 , 931 , 1225 , 1519 , . . . M 51 , 150 : 2601 , 10251 , 17901 , 25551 , 33201 , 40401 , . . . M 25 , 200 : 625 , 5625 , 10625 , 15625 , 20625 , 25625 , . . . M 341 , 620 : 116281 , 327701 , 539121 , 750541 , 923521 , 961961 , . . . Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 4 / 8

  17. 1000 500 1500 2000 2500 reducible irreducible Periodicity in ACMs Question [Baginski–Chapman, 2014] When is the list of irreducibles in M a , b (eventually) periodic? Use IrreduciblesUpToElement() to precompute reducible elements: M 1 , 4 : 1 , 25 , 45 , 65 , 81 , 85 , . . . → M 5 , 20 : 25 , 125 , 225 , 325 , 425 , 525 , . . . → M 7 , 42 : 49 , 343 , 637 , 931 , 1225 , 1519 , . . . M 51 , 150 : 2601 , 10251 , 17901 , 25551 , 33201 , 40401 , . . . → M 25 , 200 : 625 , 5625 , 10625 , 15625 , 20625 , 25625 , . . . M 341 , 620 : 116281 , 327701 , 539121 , 750541 , 923521 , 961961 , . . . Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 4 / 8

  18. 1000 500 1500 2000 2500 reducible irreducible Periodicity in ACMs Question [Baginski–Chapman, 2014] When is the list of irreducibles in M a , b (eventually) periodic? Use IrreduciblesUpToElement() to precompute reducible elements: M 1 , 4 : 1 , 25 , 45 , 65 , 81 , 85 , . . . → M 5 , 20 : 25 , 125 , 225 , 325 , 425 , 525 , . . . → M 7 , 42 : 49 , 343 , 637 , 931 , 1225 , 1519 , . . . M 51 , 150 : 2601 , 10251 , 17901 , 25551 , 33201 , 40401 , . . . → M 25 , 200 : 625 , 5625 , 10625 , 15625 , 20625 , 25625 , . . . M 341 , 620 : 116281 , 327701 , 539121 , 750541 , 923521 , 961961 , . . . M 7 , 42 : Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 4 / 8

  19. The periodic case Question [Baginski–Chapman, 2014] When is the list of irreducibles in M a , b (eventually) periodic? Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 5 / 8

  20. The periodic case Question [Baginski–Chapman, 2014] When is the list of irreducibles in M a , b (eventually) periodic? Theorem If a | b and a > 1 , then M a , b has periodic irreducible set. Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 5 / 8

  21. The periodic case Question [Baginski–Chapman, 2014] When is the list of irreducibles in M a , b (eventually) periodic? Theorem If a | b and a > 1 , then M a , b has periodic irreducible set. Example M 5 , 20 = { 5 , 25 , 45 , 65 , 85 , 105 , 125 , 145 , 165 , 185 , 205 , 225 , 245 , . . . } Reducible elements: 25 = 5 · 5 525 = 5 · 105 1025 = 5 · 205 125 = 5 · 25 625 = 5 · 125 1125 = 5 · 225 225 = 5 · 45 725 = 5 · 145 1225 = 5 · 245 325 = 5 · 65 825 = 5 · 165 1325 = 5 · 265 425 = 5 · 85 925 = 5 · 185 1425 = 5 · 285 Christopher O’Neill (UC Davis) Periodicity of irreducibles in ACMs Jan 6, 2017 5 / 8

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