On the Number of Carries Occurring in an Addition Mod
-
Jean-Pierre Flori
and
Hugues Randriam
Abstract.
In this paper we study the number of carries occurring while performing an addition modulo 
.
For a fixed modular integer t, it is natural to expect the number of carries occurring when adding a random modular integer a to be roughly the Hamming weight of t. Here we are interested in the number of modular integers in 
producing strictly more than this number of carries when added to a fixed modular integer 
. In particular it is conjectured that less than half of them do so. An equivalent conjecture was proposed by Tu and Deng in a different context.
Although quite innocent, this conjecture has resisted different attempts of proof
and only a few cases have been proved so far.
The most manageable cases involve modular integers t whose bits equal to 
are sparse.
In this paper we continue to investigate the properties of 
, the fraction of modular integers a to enumerate, for t in this class of integers.
Doing so we prove that has a polynomial expression and describe a closed form for this expression. This is of particular interest for computing the function giving and studying it analytically.
Finally, we bring to light additional properties of in an asymptotic setting and give closed-form expressions for its asymptotic values.
© 2012 by Walter de Gruyter Berlin Boston
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Articles in the same Issue
- Masthead
- On Weakly Complete Sequences Formed by the Greedy Algorithm
- Bijective Proofs of Vajda's Ninetieth Fibonacci Number Identity and Related Identities
- Newman Polynomials, Reducibility, and Roots on the Unit Circle
- A Property of Twin Primes
- Number of Permutations with Prescribed Up-Down Structure as a Function of Two Variables
- A -Expansion Associated to Sturmian Sequences
- On the Number of Carries Occurring in an Addition Mod
- A Probabilistic Look at Series Involving Euler's Totient Function
- Fibonacci Variations of a Conjecture of Polignac
- Extending Nathanson Heights to Arbitrary Finite Fields
- On the Maximal Cross Number of Unique Factorization Zero-Sum Sequences over a Finite Abelian Group
- An Explicit Bound for Aliquot Cycles of Repdigits
- A Note on the Minimal Number of Representations in
- A Unified Proof of Two Classical Theorems on CNS Polynomials
- On the Problem of Molluzzo for the Modulus 4
- On Multiplicative Functions with Bounded Partial Sums
- Avoiding Type or Patterns in a Partition of a Set
