A recursive definition of $p$-ary addition without carry

F

RANÇOIS

L

AUBIE

A recursive definition of p-ary addition without carry

Journal de Théorie des Nombres de Bordeaux, tome

11, n

o

_{2 (1999),}

p. 307-315

<http://www.numdam.org/item?id=JTNB_1999__11_2_307_0>

© Université Bordeaux 1, 1999, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

A

recursive

definition

of

_p-ary

addition without

carry

par

FRANÇOIS

LAUBIE

RÉSUMÉ. Soit p un nombre premier. Nous montrons dans cet

article que l’addition en base _p sans retenue

_possède

une définition

récursive à l’instar des cas _{où p =} 2 et p = 3 _qui étaient

_deja

connus.

ABSTRACT. Let p be a

_prime

number. In this _paper we _prove that the addition in p-ary without carry admits a recursive definition like in the

_already

known cases p = 2 and _p = 3.

1. INTRODUCTION

Let _p be a

_prime

number. For _any two natural

_integers

a and

b,

let us

denote

_by

a

_+p

b the natural

integer

obtained

writing

a and b in p-ary and

then

_adding

them without _carry.

In the case where _p =

2,

this

operation

called

nim-addition,

plays

a crucial

role in the

_theory

of some _games

[1]

and in the

theory

of

lexicographic

codes

of Levenstein

_[6],

_Conway

and Sloane

_[2].

The _map

_{(a, b)}

H a ₊₂ b is the

Grundy

function of the directed

_graph

whose vertices are the

_pairs

(a, b)

of natural

_integers

and arcs the

_pairs

of vertices

((a’,

b’),

(a, b))

such that

either a’ a and b’ = b _or a~’ = _a and b’ b. Therefore the nim-addition

can be defined

_recursively

as follows:

Thus the nim-addition is the first

_regular

law on N in the sense

_{that, given}

all a’ +2 b and a ₊₂ b’ with a’ a and b’

b,

a +2 b is the smallest natural

integer

which is not excluded

_by

the rule:

Surprisingly,

it is a _group law on N.

For _any

_prime

number

_{p &#x3E; 3,}

the addition

_+p

takes

_place

in the

_theory

of some

_{generalized nim-games}

~7~, [8]

and also in the

_theory

of some

greedy

codes

_[4].

Moreover this addition

_plays

a crucial role in the recent

deter-mination of the least

_possible

size of the sumset of two subsets of

and he asked the

_question

if such a recursive definition exists for

_-I-p

when-ever _p is a

_prime

number.

The aim of this _{paper is} to answer

positively.

This answer

provides

us

with a definition "a la

Conway"

of

prime

numbers.

2. THE

_{-I-p-ADDITION}

TABLE AS A GRAPH

Let

_lFp

be the finite field with _p

_elements;

for A E

_Fp,

let A

be the

representative

number of the class A

_belonging

to

_{{0,1, - " p -1~}

_and,

for

a E

N,

define

_A.pa=

_{a -p a =}

_{a +p a +p w +p a}

with a

terms a.

The

_{operations +p}

and _-p

_{provide N}

with a structure of

_Fp-vector

_space

isomorphic

to the

_IFp-vector

space of

polynomials

We define a directed

graph!gp

as follows:

- the set of its vertices is N x

N,

- the

arcs of

_gp

are the

pairs

of vertices

( (a’, b’ ),

(a, b) )

such that

-aa

6’ 6,

-

a - a, -,

- a’

_{= b +p ( 1- a)}

_{-p r for} some r E

_{and A e Fp .}

The

_graph

_gp

does not admit

circuit;

thus the

_Grundy

function of

_~p

is

the

_unique

_{map g} of N x N-N such that:

Proposition

1. The

_Grundy

_{function of}

_Gp

is the addition map:

(a, b)

H

a +p b.

First of

_all,

we

_give

some lemmas on the natural

ordering

of the

repre-sentative set

_{0,1,’"

_{p -}

_1}

of

_{Fp .}

It is sometimes more convenient to express them in terms of the

_{following ordering}

on

_Fp:

where ü

_{(resp. v)}

is the

_{representative}

number of u E

_{Fp (resp. v}

E

_Fp)

belonging

to

_{{0,1," - ip -}

1}-Lemma 1. For all _{u, v E}

_Fp

2

Thus

Lemma 2. Let _{u, r, s} be elements

_of

_Fp

such that r ~ s and u + r - u.

Lemma 3. Let _{u, v,} r be elements

_{o, f}

_IFp

such that u - u + r, v - v + r and

u --i- v -~- r ~ u -f- v. Then there

_{E IFp}

such

_{that s -~- t = r,}

u + s -~ u

and v + t - v.

Proof.

Conditions:

are

_equivalent

to:

Since

v

&#x3E; _{p -} r with f _{p -} 1 -

_n,

we have

v

Hence &#x3E;

_max(fi,,b)

+ 1. Moreover

_{n+~2013p~ ~T~.}

_{Therefore, by}

Lemma

_l,M+~p20131

and the conditions

_(C)

are

_equivalent

to:

or, more

_simply,

to:

_{-f -1 p - r u -f - v .}

We are

looking

for s and t E

_I~p

such that:

or

_equivalently

such that:

with _p =

_{p - f, u}

=

p - s

and _T =

p - t.

&#x3E;bbom the condition

v )

+

_v,

it is clear that such

_{integers a}

and r do exist. Thus the lemma

is

_proved.

0

Now,

for _any natural

_integer

x, let a? be its class modulo p, let a~ =

Ei&#x3E;O Xipi

with Xi E

~0,1, ~ ~ ~ ,

p -

11

its p-ary

expansion,

and let

iz

be

the largest

index i &#x3E; 0 such that 0. In order to summarize all these

Proof of Proposition

1. Let _a, b be natural

_integers.

For any natural

integer

c a

_{+p b,}

there exists r E N* so that c = a

_+p

b

_+p

r. We will _prove that

for _any r E N* such that a

_+p

b

_+p

r a

_{+p b,}

there exists A E

_Fp

such that

a

+p A .p

r a and b

_+p

(1 - A)

_.p r b. With the notations of Lemma

_4,

we have:

There exist

_{s, t}

E

_Fp

such that _q - _s

_{+ t,}

as,. + s -~

az,, and

bzr

-f - t ~ Let A = _{s E}

IFp ;

then: s =

Aq, t

=

( 1 -- ~ ) rZr ,

_q

_{+ «} air and

+ ( 1 - ^)TZr ~

bz,. ;

in other words :

_{a +p A .p r}

a and v

(Lemma 4).

Therefore a

_+p

b =

min(NBEp)

where

Ep

is the set of all the natural

integers a’

+p

b’ with a’ a, b’ b and such that there exist A E

_1Fp

and

r E N*

_{satisfying a’ =}

a

_+p

A _{-p r,} b’ = b

+p

( 1 -

A)

-p r. This means that

(a, b)

H a

_+p

b is the

_Grundy

function of

_{gp .}

0

Corollary.

(S.

Eliahou,

M. Kervaire

_{[3]) -}

Let us denote

_by

[0, a]

the

inter-val

_~a’

_{a~ ,}

_for

a E Then

_for

all a, b E ~1 there exists c a + b such that

_{~0, a~}

_+p

_{[0, b]}

=

[0, c~ .

Proof.

Let c =

max([O,

a]

+p

[0, b])

and let a,

bl

:5

b such that c =

al

_{+p bl.}

For all d c there exist A E

_IFp

and r E N* such that d = A _{.p a,}

_+p

_{( 1 - A)}

_-p

_bi,

A _-p ai al and

( 1 -

A)

_p

bl

bl ;

therefore

d E ~~ ~ a~ +p ~~ ~ b~ -

D

Remark. With the notations of the

_proof

of

_Proposition

_1,

we have:

1.

_E2

_{= {a}

₊₂

_{b’, a’}

₊₂

_{b ;}

a’

_{a, b’}

_&#x26;},

2.

_{E3 -}

_{a

₊₃

_{b’, a’}

₊₃

_{b ; a’}

_{a, b’}

_b~

U

_fall

₊₃

_{b", a"}

_{a, b"}

b,

a _{+3 b"} = a" +3

6}

because in this _case, A = 0 or A = 1 or A = 1 - A. 3. In the case where

_{p &#x3E; 5,}

the situation is a little more

complicated

because the formula a

_+p

b = will be

effectively

recursive

,

only

when we can describe the set

Ep

using only

pairs

(a, #)

E N x N

with a _a,

_,Ci

b and

_(a,,C3)

_#

_(a,

_b).

3. A RECURSIVE EXCLUSION ALGORITHM FOR a

_+p

b

Given a

_prime

number _p and a

_pair

(a,

b)

of natural

_integers,

we will

integers

of the kind a’

_{+p b’}

_~

a

_+p

b with a’ a and b’ b without

_using

any

pair

of

integers

(a", b")

such that a" &#x3E; a or b" &#x3E; b.

For all S C

_N,

S* means

S)(0).

Let J~I

and N

be two finite sets of natural

_integers

such that

n N

=

~0,1~

and let and

_(bn)nEN

be two _sequences of natural

_integers

(respectively

indexed

_by

,Nl and

_JU)

_satisfying

the conditions:

-ao=a~bo=b~

-

al

+p b = a +p bl7

-

V(m,n)

E M* x

N*,

_{am a,}

bn

_b,

- dm E E J~I * such that and

am

_+p

b =

ak

_+p

bm-k ~

&#x3E;

- Vn E E

N*

such and _a

+p bn

=

+p bl.

Such a

_pair

of _sequences

(bn)nEN)

is called a

_p-chain

of

(a, b)

of

_length

card J1~! * + card

N* .

Remark. 1- The

_p-chains

of

_{(a, b)}

of

_length

2 are the

pairs

lb,

with a, a,

b1

b and a

_{+p bl}

= _a,

+p

b

(see

the formula of S. Norton in

the

_{introduction).}

2 - For a

_p-chain

of

(a, b)

of

_length

&#x3E;

_3,

we have _a2

_+p

b = _al

+p b,

or a

_{+p b2 =}

_a,

_{+p bi,}

_a3

_+p

b =

a2

_{+p bl}

provided

that

(a2,

bl )

lies in the

p-chain,

or a3

_+p

b = _al

+p b2 provided

that

(a1, b2)

lies in the

_p-chain.

For convenience we extend our definition to

_length

1

_p-chain

of

_{(a, b):}

it is the

_pairs

_(a,

or

(la, a, 1, b)

with al a,

b¡

b.

A

_p-chain

_{(a, b)}

is called a

p-exclusion

chain for a

_+p

b

(or

of

(a,

b))

if V n E

_{U N*,}

Finally

the set of all

_integers

a’

_+p

b’ where

_{(a’, b’)}

_belongs

to any

p-exclusion chain for a

_+p

b of

length

_{p -1}

is called the

p-exclusion

set for

a

_+p

b

(or

of

(a, b));

it’s denoted

by

Ep(a,

b).

We will _prove:

Theorem.

_{((a’, b’),}

_{(a, b))}

is an arc

of

gp

if

and

only if

there exists a

p-exclusion chain

_for

a

_+p

b

of

p -1 containing

(a’,

b’).

In other

words : a

_+p

b =

b)).

Lemma 5. Let

_{(bn)nEN) be}

a

p-chain of

(a, b)

of length &#x3E;

2.

There exists r E ~1* such that am = _a

+p

rn _.p r and

bn

= b

+p

n _.p r

for

all

(m,

n)

E x

Jlf.

Thus

_{if ptm}

+ n then am

_{+p bn}

= a

_+p

b.

Proof.

Let r E N* such that al =

a +p r;

then

a +p bi

= _al

+p b

=

a +p r +p b,

therefore

_{bl -}

b

_+p

r.

Suppose

that for _any k E and I

E N

with

1 k m - l and 1 ~ r~ - l we have _ak =

a +p k ~p r

and

bt

= then there exists

_ko

E J~l such that and

Proposition

2. Let

_{(bn)nEN) be}

a

p-chairt of

(a, b).

Let

(m, n)

E J1~ x

,N

such then

_{((am, bn), (a, b))}

is an arc

of

Qp.

Proo f . -

If the

_length

of this chain is

_1,

this is

_clear;

if

_not,

let r E N* such that _am =

a+pm~pr

and

_bn

=

b+pn~pr (Lemma

5).

Let IA

be the class modulo _p of m + n. then _a,~ = _a

(Lemma 5)

and

((a~, bn), (a, b))

is

an arc of

_gp

since

bn

b.

_{If pf m}

and + n, let A E

_{Fp (A 7~ 0, 1)}

be the class modulo p of

~+n

then m _{-p r}

_{= A -p ~}

and n _-P r =

(1- A)

-p s where

s _{= p .p} r. Thus

( (a~,,

bn ), (a,

b) )

is an arc of C7

We

_{just proved}

that if

_{((am)mEM, (bn)nEN)}

is a

p-exclusion

chain for a

_+p

b

then,

for _every

(m, n)

E x

N*,

7

((am, bn), (a, b))

is an arc of

Gp.

Now,

in order to prove the converse, we will describe an

_{algorithm looking}

like the Euclid

_algorithm

for the

_gcd.

Let _uo, such that

_{uo 0}

vo. Define vi E

_IF#

as follows:

. - ... ,

Lemma 6. There is an

_integer

N _{p - 2} such that uN = _VN.

Proof.

If

_{un 0 vn}

then _un+1 + =

min(un,

i vn);

moreover if un « _Vn

then un « un - vn =

(Lemma

2)

and un

vn+1(=

vn);

therefore

min(un,vn) - mk(un+i, vn+i)

and the _sequence is

_strictly

increasing

as

_long

as _vn-,. Thus:

min( 11,0, vo)

-

_{min(u1, vl)}

-~ ... ~

_min(UN-1,

UN = _VN where N = 1 +

maxf k E

N

Ul, 34

Finally

N _{p -} 2 because

min( uo, vo) =1=

0. 0

Let w = _UN = _{vN E}

K

and define two

_increasing

_sequences of natural

integers

and follows:

1

- if

vN-n then + vn and vn,

- if

uN-n then and vn+1 = Jln + vn.

Setting

M 1 ~

N +

_{1 ~}

we

_get

_by

iteration:

Lemma 7.

_{V p}

E

_3p,’

E

V v E 3v’ E

~ ~ - ~

E M

Proof.

= _{uN, vlzu} =

VN and for 1 n

N,

we have either _{uN-n =}

+ and VN-n = VN-n+17 or and

uN-n+l + · The lemma follows

by

recurrence. 0

Lemma 9. For 1 n N

_{+ 1,}

_ptiln

Proof.

Obvious

_by

the

_preceding

lemma. D

Lemma 10. CardM* +

CardN*

Proof.

For 1 n

_N7Un+Vn

= therefore the _sequence

((Itn

is

_{strictly decreasing}

in for the

_ordering

-~. Moreover

CardM* +

CardN*

=

Card(fwl

_{U +}

vn)w ;

1 _n

NJ).

D Now we can

complete

the

Proof of

the theorem. Let

_{( (a’, b’),}

_(a,

_{b) )}

be an arc of

_gp

with a’ = _a

+p

A _.p r

a, b’

= b

+p

(1 - A)

-p r

b,

A E

1Fp ,

r E I~’’‘ . We will construct a

p-exclusion

chain for a

_{+p b,}

_containing

(a’, b’),

of

_length

_{p -1.}

If A = 0 _or 1 there exists such _an obvious chain of

length

1.

If A

_{= 1}

7

(p ~ 3),

is such

a p-exclusion

chain of

length

2 for a

_+p

b.

Now we _suppose

_{that A #}

0,1,

and therefore that

_{p &#x3E;}

5.

_Writing

r = _p-ary, let us recall

_{that ir}

denotes the

_largest

index i

such that _rir

_~

0. Let uo -

Aq

E vo =

(1 -

A)F,7,-

E

_JB;;

then

uo + 0 and uo - 0. So we can construct as above the _sequences

with _w, the

_increasing

_sequences of

integers

with P,1 =

v1 - 1 and their associated

1nN-~-1~.

Lemma 11. The

_equality

_{ILN+1(1 -}

_À)

=

vN+iA

holds in

1F’‘p .

Proof. By

Lemma

_8,

_{IIN+LW =} uo -

Àrir

and vn+lw = vo =

(1 - À)rir

with ~.u

# 0

and

_q

~

0. 0

Thus there exists a

unique

natural

integer R

such that R =

a -p

_r and _{vN+1 -p} R =

(1 -

A)

.p r.

Lemma 12.

_Rïr

= _w.

Proof.

=

up =

Xrir

0

For _every

_{(p, v)}

E M x let a~ = and

bv

=

b +p v .p R.

Lemma 13. For _every

_{(p, v)}

E x

.IV*,

_a~ a and

_bp.

b.

Now

_{clearly a p-chain}

of

_{(a, b) (Lemmas 7}

and

_13),

containing

(a’, b’) (Lemma 11),

of

_length

_{p -} 1

_{(Lemma 10),}

which is a

p-exclusion

chain for a

_+p

b

(Lemma

9).

Remark. 1. In the cases where _{p = 2} or

_3,

_every

_p-chain

of

_{(a, b)}

of

length

p - 1 is a

p-exclusion

chain for a

_+p

b.

2. In the case where _{p =}

_5,

a 5-chain of

length

4 is not

_necessarily

a

5-exclusion

_chain;

we can however write a

complete

readable formula

of the same kind as Norton’s formula for _p = 3: let _a, b _E

N;

a’, a",

a"’

(resp.

b’, b",

b"’)

are variables

_taking

their values in

{0,1, -",

a -

1}

(resp.

{0,1, -" ,

b -

_1);

let us consider the sets:

r . 1

Then a ₊₅ b = U

S2

U

S3

U

S4).

3. Given a natural

_{integer v &#x3E;}

2 not

necessarily

prime

and two natural numbers _a,

_b,

let us

generalize

the definition of the

p-exclusion

set

Ep(a,

b)

of

_{(a, b)}

_replacing

_p

_by

v in the

previous

definition.

Thus a v-exclusion chain

(bn)nEN)

of

(a,b)

is of

length

v - 1 and such that V m E

_,M*,

V n E

N*, vf m

and Then

setting

a *, b =

min(NBEv (a,

b)),

_*v is a _group law on N if and

_only

if

v is a

_prime

number.

Proof.

In fact if v is a

composite

number then _*v is not an associative

law. Let d be a _proper divisor of _v; the

following equalities

hold:

exclusion chain

_oflength

Therefore:

_{((v-d)*v(d-1))*"1}

= 0 and:

_{(v - d)}

_*"

_{((d - 1)}

_*w

₁₎

= _{v. 0}

4. If we

_replace

in the definition of _*£1 the

_previous

conditions

_{(m, n)}

E

M* x

N*

_{v t m}

and

_{by v}

is

_relatively

_prime

to m and _n, then we

_get

_{*v =}

_+p

where _p is the smallest

_prime

divisor of v.

Acknowledgments.

I thank T. Moreno and P.

_Segalas,

students at the

University

of

_Limoges,

for

_testing

several

_algorithms

in Turbo Pascal on a

PC.

REFERENCES

[1] C. L. Bouton, Nim, a _{game with} a _complete mathematical theory. Ann. Math. Princeton 3

(1902), 35-39.

[2] J. H. _Conway, N.J.A. _{Sloane, Lexicographic} Codes, Error Corrrecting Codes from Game

The-ory. IEEE Trans. Inform. Theory 32 _(1986), 337-348.

[3] S. _Eliahou, M. _Kervaire, Sumsets in vector spaces over _{finite fields.} J. Number Theory 71

(1998), 12-39.

[4] F. _Laubie, On linear greedy codes. to appear.

[5] H. W. _Lenstra, Nim Multiplication. Séminaire de Théorie des Nombres de Bordeaux _1977-78,

exposé 11, (1978).

[6] V. _Levenstein, A class of Systematic Codes. Soviet Math Dokl. 1 _(1960), 368-371.

[7] S. Y. R. _{Li, N-person} Nim and N-person Moore’s Games. Int. J. Game _{Theory 7 (1978),}

31-36.

[8] E. H. Moore, A _{generalization of} the Game called Nim. Ann. Math. Princeton 11 _(1910),

93-94.

Francois LAUBIE

UPRESA CNRS 6090 et INRIA de _Rocquencourt

Departement de _{Mathematiques} Facult6 des Sciences de _Limoges

123, avenue Albert Thomas

87060 LIMOGES Cedex E-mail : laubieihmilim. f r