Additive Number Theory via Automata Theory

Rajasekaran, Aayush; Shallit, Jeffrey; Smith, Tim

doi:10.1007/s00224-019-09929-9

Cited by 17 publications

(16 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, Cilleruelo et al [4], improving a result of Banks [1], proved that, for g ≥ 5, the set of natural numbers whose base-g representations are palindromes is an additive basis of order 3. Rajasekaran et al [18] showed that the same is true for g = 3, 4 but C. Sanna [2] not for g = 2; and they proved that the binary palindromes are an additive basis of order 4. Moreover, Madhusudan et al [15] proved that the set of natural numbers whose binary representations consist of two identical repeated blocks is an asymptotic basis of order 4, while Kane et al [14] gave a generalisation regarding k repeated blocks.…”

Section: Introductionmentioning

confidence: 93%

Additive Bases and Niven Numbers

Sanna

2021

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

Let $g \geq 2$ be an integer. A natural number is said to be a base-g Niven number if it is divisible by the sum of its base-g digits. Assuming Hooley’s Riemann hypothesis, we prove that the set of base-g Niven numbers is an additive basis, that is, there exists a positive integer $C_g$ such that every natural number is the sum of at most $C_g$ base-g Niven numbers.

show abstract

Section: Introductionmentioning

confidence: 93%

Additive Bases and Niven Numbers

Sanna

2021

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Observe that both sequences obey the gcd condition, because gcd(e i , e i+1 , e i+2 ) = gcd(o i , o i+1 , o i+2 ) = 1 for all i ≥ 0. Additive properties of these numbers were studied previously in [13,Thm. 2].…”

Section: The Odious and Evil Numbersmentioning

confidence: 99%

Frobenius Numbers and Automatic Sequences

Shallit¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The Frobenius number g(S) of a set S of non-negative integers with gcd 1 is the largest integer not expressible as a linear combination of elements of S. Given a sequence s = (s i ) i≥0 , we can define the associated sequence G s (i) = g({s i , s i+1 , . . .}). In this paper we compute G s (i) for some classical automatic sequences: the evil numbers, the odious numbers, and the lower and upper Wythoff sequences. In contrast with the usual methods, our proofs are based largely on automata theory and logic. 2010 AMS MSC Classifications: 11D07, 11B85, 11B83.

show abstract

“…For example, Cilleruelo, Luca, and Baxter [4], improving a result of Banks [1], proved that, for g ≥ 5, the set of natural numbers whose base-g representations are palindrome is an additive basis of order 3. Rajasekaran, Shallit, and Smith [18] showed that the same is true for g = 3, 4 but not for g = 2; and they proved that the binary palindromes are an additive basis of order 4. Moreover, Madhusudan, Nowotka, Rajasekaran, and Shallit [15] proved that the set of natural numbers whose binary representations consist of two identical repeated blocks is an asymptotic basis of order 4, while Kane, Sanna, and Shallit [14] gave a generalization regarding k repeated blocks.…”

Section: Introductionmentioning

confidence: 98%