pdf

data

On a conjecture of Rudin on squares in arithmetic progressions

Enrique González-Jiménez and Xavier Xarles

Data (Magma and Sage files)

Section 5: Five squares in arithmetic progressions: the technique

• We have implemented in Magma v2.18-8 the method described in section 5. This methods tries to compute all the rational points of the genus 5 curve attached to a 5-tuple: rationalpoints5tuples.mg (Magma)
  

Section 5.1: The algorithm at work

     OUTPUTs of the examples at section 5.1 using the functions implemented in Magma related to Section 5:

• { 0, 1, 2, 4, 7 }
• { 0, 1, 2, 5, 7 }
• { 0, 1, 3, 7, 8 }
• { 0, 1, 4, 7, 8 } and over Q
• { 0, 3, 5, 6, 10 }
• { 0, 2, 4, 5, 11 }
• { 0, 13, 24, 33, 49 }
  

Section 6: Summary of the Computations

• Functions related to subsets of positive integers: Rudin.sage (Sage)
  
  
• The following files contain the output related to the computations of all the 5-tuples where our algorithm has finished:
  

  5tuples_output.m

  5tuples_output_extended.m

  
  
• Resume:
  
  

  #	Positive integers nI associated to equivalences classes of the 5-tuples I of {0,...,51} such that	file
  111338	all the elliptic quotient of the genus 5 curves CI have only the 8 trivial points	Ms_52.sage
  424	the implementation has obtained all the rational points attached to the corresponding genus 5 curve CI and #CI(Q)>16	Ms_true_NO_trivial_52.sage
  26165	the implementation has obtained all the rational points attached to the corresponding genus 5 curve CI and #CI(Q)=16	Ms_true_trivial_52.sage
  83716	the algorithm does not compute CI(Q) but finishes	Ms_false_52.sage
  1033	the algorithm does not finish	Ms_No_Finish_52.sage

Section 7: Consequences and comments

• For k=5,...,10 the list Tk contains the sorted list of n_I associated to primitive k-tuples I of {0,...,51}such that we have not been able to compute C_I(Q)

  T5

  T6

  T7

  T8

  T9

  T10

  the above lists have been obtained using TTs.sage.
  
  
• For k=5,...,12 the list TBk.sage contains the sorted list of n_I associated to primitive k-tuples I of {0,...,51}such that we have not been able to compute C_I(Q) or C_I(Q) has more than the 16 trivial points:
  

  TB5

  TB6

  TB7

  TB8

  TB9

  TB10

  TB11

  TB12

  the above lists have been obtained using TTBs.sage.
  
  
• The above lists have been used to prove the Strong and Super-Strong Rudin's Conjectures, and in particular the Table 1 and Table 2:
  

  table1_Conjecture_S.out

  table2_Conjecture_SS.out