Report on D ( 1)-quadruples Prologue Terminology Main problems - PowerPoint PPT Presentation

D ( − 1)- quadruples Mihai Cipu Report on D ( − 1)-quadruples Prologue Terminology Main problems Existence A conjecture Mihai Cipu Classical approach to find quadi Starting point IMAR, Bucharest, ROMANIA Alternative approach Representation Theory XVI Different viewpoint Dubrovnik, 28 th June 2019 A particular case The general case (joint work with N. C. Bonciocat and M. Mignotte) References

Outline D ( − 1)- 1 Prologue quadruples Mihai Cipu Terminology Prologue 2 Main problems Terminology Existence Main problems Existence A conjecture A conjecture Classical 3 Classical approach to find quadi approach to find quadi Starting point Starting point Alternative approach 4 Alternative approach Different viewpoint Different viewpoint A particular case The general case A particular case References The general case 5 References

Terminology D ( − 1)- quadruples D ( n ) − m -set = set of m positive integers, the product Mihai Cipu of any two being a perfect square minus n Prologue Terminology Main problems Existence A conjecture Classical approach to find quadi Starting point Alternative approach Different viewpoint A particular case The general case References

Terminology D ( − 1)- quadruples D ( n ) − m -set = set of m positive integers, the product Mihai Cipu of any two being a perfect square minus n Prologue Terminology Main problems n -pair = D ( n ) − 2-set Existence A conjecture Classical approach to find quadi Starting point Alternative approach Different viewpoint A particular case The general case References

Terminology D ( − 1)- quadruples D ( n ) − m -set = set of m positive integers, the product Mihai Cipu of any two being a perfect square minus n Prologue Terminology Main problems n -pair = D ( n ) − 2-set Existence A conjecture Classical n -triple = D ( n ) − 3-set approach to find quadi Starting point Alternative approach Different viewpoint A particular case The general case References

Terminology D ( − 1)- quadruples D ( n ) − m -set = set of m positive integers, the product Mihai Cipu of any two being a perfect square minus n Prologue Terminology Main problems n -pair = D ( n ) − 2-set Existence A conjecture Classical n -triple = D ( n ) − 3-set approach to find quadi Starting point n -quadruple = D ( n ) − 4-set Alternative approach Different viewpoint A particular case The general case References

Terminology D ( − 1)- quadruples D ( n ) − m -set = set of m positive integers, the product Mihai Cipu of any two being a perfect square minus n Prologue Terminology Main problems n -pair = D ( n ) − 2-set Existence A conjecture Classical n -triple = D ( n ) − 3-set approach to find quadi Starting point n -quadruple = D ( n ) − 4-set Alternative approach Different viewpoint n -quintuple = D ( n ) − 5-set A particular case The general case References

Terminology D ( − 1)- quadruples D ( n ) − m -set = set of m positive integers, the product Mihai Cipu of any two being a perfect square minus n Prologue Terminology Main problems n -pair = D ( n ) − 2-set Existence A conjecture Classical n -triple = D ( n ) − 3-set approach to find quadi Starting point n -quadruple = D ( n ) − 4-set Alternative approach Different viewpoint n -quintuple = D ( n ) − 5-set A particular case The general case References n = − 1 = ⇒ pardi, tridi, quadi

Fundamental question D ( − 1)- quadruples Mihai Cipu How large a D ( n ) − m -set can be? Prologue Terminology Main problems Existence A conjecture Classical approach to find quadi Starting point Alternative approach Different viewpoint A particular case The general case References

Fundamental question D ( − 1)- quadruples Mihai Cipu How large a D ( n ) − m -set can be? Prologue Terminology Infinite if n = 0 Main problems Existence A conjecture Classical approach to find quadi Starting point Alternative approach Different viewpoint A particular case The general case References

Fundamental question D ( − 1)- quadruples Mihai Cipu How large a D ( n ) − m -set can be? Prologue Terminology Infinite if n = 0 Main problems Existence A conjecture From now on n � = 0 Classical approach to find quadi Starting point Alternative approach Different viewpoint A particular case The general case References

Fundamental question D ( − 1)- quadruples Mihai Cipu How large a D ( n ) − m -set can be? Prologue Terminology Infinite if n = 0 Main problems Existence A conjecture From now on n � = 0 Classical approach to find quadi Starting point Dujella 2004 Any D ( n ) − m -set has Alternative approach Different m ≤ 31 if 1 ≤ | n | ≤ 400 viewpoint A particular case m < 15 . 476 log | n | if | n | > 400 The general case References

Better answers D ( − 1)- quadruples Brown, Gupta-Singh, Mohanty-Ramasamy 1985 Mihai Cipu There are no D (4 k + 2)-quadruples Prologue Terminology Main problems Existence A conjecture Classical approach to find quadi Starting point Alternative approach Different viewpoint A particular case The general case References

Better answers D ( − 1)- quadruples Brown, Gupta-Singh, Mohanty-Ramasamy 1985 Mihai Cipu There are no D (4 k + 2)-quadruples Prologue Dujella 2004 There are no D (1)-sextuples and only Terminology Main problems finitely many D (1)-quintuples Existence A conjecture Classical approach to find quadi Starting point Alternative approach Different viewpoint A particular case The general case References

Better answers D ( − 1)- quadruples Brown, Gupta-Singh, Mohanty-Ramasamy 1985 Mihai Cipu There are no D (4 k + 2)-quadruples Prologue Dujella 2004 There are no D (1)-sextuples and only Terminology Main problems finitely many D (1)-quintuples Existence A conjecture Dujella-Luca 2005 Any D ( n ) − m -set with n prime has Classical approach to m < 3 · 2 168 find quadi Starting point Alternative approach Different viewpoint A particular case The general case References

Better answers D ( − 1)- quadruples Brown, Gupta-Singh, Mohanty-Ramasamy 1985 Mihai Cipu There are no D (4 k + 2)-quadruples Prologue Dujella 2004 There are no D (1)-sextuples and only Terminology Main problems finitely many D (1)-quintuples Existence A conjecture Dujella-Luca 2005 Any D ( n ) − m -set with n prime has Classical approach to m < 3 · 2 168 find quadi Starting point Alternative Dujella-Fuchs 2005 There is no D ( − 1)-quadruple approach Different whose smallest element is ≥ 2. Hence, there is no viewpoint A particular case D ( − 1)-quintuple The general case References

Better answers D ( − 1)- quadruples Brown, Gupta-Singh, Mohanty-Ramasamy 1985 Mihai Cipu There are no D (4 k + 2)-quadruples Prologue Dujella 2004 There are no D (1)-sextuples and only Terminology Main problems finitely many D (1)-quintuples Existence A conjecture Dujella-Luca 2005 Any D ( n ) − m -set with n prime has Classical approach to m < 3 · 2 168 find quadi Starting point Alternative Dujella-Fuchs 2005 There is no D ( − 1)-quadruple approach Different whose smallest element is ≥ 2. Hence, there is no viewpoint A particular case D ( − 1)-quintuple The general case References Dujella-Filipin-Fuchs 2007 There are only finitely many D ( − 1)-quadruples

Very recent results D ( − 1)- quadruples Mihai Cipu Prologue Terminology Main problems He-Togb´ e-Ziegler 2019 There is no D (1)-quintuple Existence A conjecture Classical approach to find quadi Starting point Alternative approach Different viewpoint A particular case The general case References

Very recent results D ( − 1)- quadruples Mihai Cipu Prologue Terminology Main problems He-Togb´ e-Ziegler 2019 There is no D (1)-quintuple Existence A conjecture Classical Bliznac Trebjeˇ sanin-Filipin 2019 There is no approach to find quadi D (4)-quintuple Starting point Alternative approach Different viewpoint A particular case The general case References

A conjecture D ( − 1)- quadruples Mihai Cipu Prologue Terminology Main problems Dujella 1993 if n �∈ S := {− 4 , − 3 , − 1 , 3 , 5 , 12 , 20 } and Existence A conjecture n � = 4 k + 2 then there exists at least one D ( n )-quadruple Classical approach to find quadi Starting point Alternative approach Different viewpoint A particular case The general case References

A conjecture D ( − 1)- quadruples Mihai Cipu Prologue Terminology Main problems Dujella 1993 if n �∈ S := {− 4 , − 3 , − 1 , 3 , 5 , 12 , 20 } and Existence A conjecture n � = 4 k + 2 then there exists at least one D ( n )-quadruple Classical approach to find quadi Conjecture for n ∈ S does not exist D ( n )-quadruples Starting point Alternative approach Different viewpoint A particular case The general case References

How to find D ( − 1)-sets D ( − 1)- quadruples Mihai Cipu Prolongation: start with a pair, extend it to a triple, then to a quadruple . . . Prologue Terminology Main problems Existence A conjecture Classical approach to find quadi Starting point Alternative approach Different viewpoint A particular case The general case References