Page 648 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 648
NUMBER THEORY
Generalization of proofs and codification of graph families
Lorenzo Sauras Altuzarra, lorenzo.sauras@tuwien.ac.at
TU Wien, Austria
In this talk, I will explain how to apply Baaz’s generalization method and a recent graph-
theoretical technique to, respectively, formal proofs from elementary number theory and certain
increasing families of simple graphs. These procedures have resulted to have a considerable po-
tential in revealing arithmetical patterns, that got reflected in theorems (for example, sufficient
conditions for a value to be a divisor of an arbitrary Fermat number) and conjectures (mainly
about integer sequences).
Frobenius number of relatively prime three Lucas numbers
Boonrod Yuttanan, boonrod.y@psu.ac.th
Department of Mathematics and Statistics, Prince of Songkla University,
Songkhla 90110, Thailand
Let a1, a2, . . . , an (n ≥ 2) be positive integers with gcd (a1, a2, . . . , an) = 1. Finding the largest
positive integer N such that the Diophantine equation
a1x1 + a2x2 + · · · + anxn = N
has no solution in non-negative integers is known as the Frobenius problem. Such the largest
positive integer N is called the Frobenius number of a1, a2, . . . , an. Various results of the Frobe-
nius number have been studied extensively, see [1]−[8]. In this talk, the Frobenius problem is
discussed in the cases n = 2 and 3. In particular, we determine the formula for the Frobenius
number of relatively prime three Lucas numbers other than results of S. Ýlhan and R. Kýper
in [8].
Keywords and phrases: Frobenius number, Lucas numbers, Fibonacci numbers.
2010 Mathematics Subject Classification: 11D07, 11B39
References
[1] A. Brauer and J.E. Shockley, On a problem of Frobenius, J. Reine Angew. Math., 211
(1962), 215−220.
[2] B.K. Gil et al., Frobenius numbers of Pythagorean triples, Int. J. Number Theory, 11 (2015),
613−619.
[3] J.M. Marín, J.L. Ramíres Alfonsín and M.P. Revuelta, On the Frobenius number of Fi-
bonacci numerical semigroups, Integers, 7 (2007) no. A14, 1−7.
[4] D.C. Ong and V. Ponomarenko, The Frobenius number of geometric sequences, Integers 8
(2008), A33, 1−3.
[5] Ö.J. Rödseth, On a linear Diophantine problem of Frobenius, J. Reine Angew. Math., 301
(1978), 171−178.
[6] E.S. Selmer and Ö. Beyer, On the linear Diophantine problem of Frobenius in three vari-
ables, J. Reine Angew. Math., 301 (1978), 161−170.
646
Generalization of proofs and codification of graph families
Lorenzo Sauras Altuzarra, lorenzo.sauras@tuwien.ac.at
TU Wien, Austria
In this talk, I will explain how to apply Baaz’s generalization method and a recent graph-
theoretical technique to, respectively, formal proofs from elementary number theory and certain
increasing families of simple graphs. These procedures have resulted to have a considerable po-
tential in revealing arithmetical patterns, that got reflected in theorems (for example, sufficient
conditions for a value to be a divisor of an arbitrary Fermat number) and conjectures (mainly
about integer sequences).
Frobenius number of relatively prime three Lucas numbers
Boonrod Yuttanan, boonrod.y@psu.ac.th
Department of Mathematics and Statistics, Prince of Songkla University,
Songkhla 90110, Thailand
Let a1, a2, . . . , an (n ≥ 2) be positive integers with gcd (a1, a2, . . . , an) = 1. Finding the largest
positive integer N such that the Diophantine equation
a1x1 + a2x2 + · · · + anxn = N
has no solution in non-negative integers is known as the Frobenius problem. Such the largest
positive integer N is called the Frobenius number of a1, a2, . . . , an. Various results of the Frobe-
nius number have been studied extensively, see [1]−[8]. In this talk, the Frobenius problem is
discussed in the cases n = 2 and 3. In particular, we determine the formula for the Frobenius
number of relatively prime three Lucas numbers other than results of S. Ýlhan and R. Kýper
in [8].
Keywords and phrases: Frobenius number, Lucas numbers, Fibonacci numbers.
2010 Mathematics Subject Classification: 11D07, 11B39
References
[1] A. Brauer and J.E. Shockley, On a problem of Frobenius, J. Reine Angew. Math., 211
(1962), 215−220.
[2] B.K. Gil et al., Frobenius numbers of Pythagorean triples, Int. J. Number Theory, 11 (2015),
613−619.
[3] J.M. Marín, J.L. Ramíres Alfonsín and M.P. Revuelta, On the Frobenius number of Fi-
bonacci numerical semigroups, Integers, 7 (2007) no. A14, 1−7.
[4] D.C. Ong and V. Ponomarenko, The Frobenius number of geometric sequences, Integers 8
(2008), A33, 1−3.
[5] Ö.J. Rödseth, On a linear Diophantine problem of Frobenius, J. Reine Angew. Math., 301
(1978), 171−178.
[6] E.S. Selmer and Ö. Beyer, On the linear Diophantine problem of Frobenius in three vari-
ables, J. Reine Angew. Math., 301 (1978), 161−170.
646
