Generalization of a theorem of Erdôs and Surânyi

ANNALES SOCIETATIS MATHEMATICAE POLONAE Séries I: COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIKI POLSK.IEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXI (1979)

Jerzy Mitek (Losice)

Generalization of a theorem of Erdôs and Surânyi

The purpose of the present paper is to prove the following

Th eo r e m. Every integer number к can be represented in the form к = + l s+2S+ ... ± m s for a suitable choice of signs, where s is a given integer ^ 2. The number of such representations is infinite.

This theorem generalizes a result stated as problem No. 5 in [2], p. 200, and quoted in [3] as problem No. 250, which deals with the case s = 2.

We need some lemmas.

For a given positive integer s we define a sequence (ask) in the following way:

П ц = — 1 , ^ 1 2 = 1 >

aik = for к ^ 2'- 1 , aik = for 2,_1 + 1 ^ к ^ 2‘ (i ^ s),

a s , k + 2 s = ü s k •

Lemma 1. We have

2 s

X askkl = 0 for f = 0, 1, . . . , s - 1.

k = 1

This lemma is well known (cf. [1], p. 50, Exercises 1 and 5).

For brevity we put As = 2Jstv-1)s!.

Lemm a 2. For an arbitrary number m and a positive integer s we have

2 s

Z ask(m + k)s = As.

k = 1

Proof. In virtue of Lemma 1 we have

2 s 2 s

X ask(m + k f = X askks.

k = 1 f c = 1

174 J. Mi tek

It is sufficient to prove that

( 1)

2

k = 1

X

It is evident that (1) is true for s = 1. Suppose that (1) is true for some s. We have

X

X

X

k =1 к = 1 к = 1

= - I as*k! + 1+ Z a* Z' (sÆ‘)2” ks + 1- "

k = l k = 1 m = 0

= - Z «лк,+ 1+ z'fm1)2” Z

= - Z ^ s+‘ + Z flrtk*+1 + 2s(s| 1) Z

= 2s( s + l ) ^ s = As+1, which completes the inductive proof of (1).

We say that an integer к is s-decomposable if there exists a positive integer m such that for a suitable choice of signs the equality

(2^{) к = ± 1S± 2}S± ... ± m s

holds.

Lemma 3. Every s-decomposable integer has infinitely many representations of the form (2).

Proof. In virtue of Lemma 2 we have

n - 2 s n - 2 s

(3) о = X

X

к = 1 k = 1

n-2s

Hence if a = ± 1 S± 2 S+ ... ± m s, then a = ± 1 S± 2 S± ... ± m s+

X

rr 2s k=l

~

X

k = i

pletes the proof.

Lemma 4. Let the number kAs + a be s-decomposable. Then the number a is s-decomposable.

Proof. For some number m and some choice of signs we have kAs + + a = ± 1 S± 2 S+ ... ± m s. Hence a = ± 1 S± 2 S± ... ± m s — kAs.

Applying к times Lemma 2, we get

k - 2 s

a = ±1S±2S± ... ± m s— X asi(m + if.

i = 1

Generalization of a theorem o f Erd'ôs and Surânyi 175

Thus the proof of Lemma 4 is completed.

P r o o f of the the ore m. Putting in (3) m = 0, n = 1, we see that the number 0 is s-decomposable. Evidently, the number 1 is s-decomposable, too.

Suppose that the number p is s-decomposable. We shall prove that p + 2 is also s-decomposable.

We have p = e x l s + e22s + ... + e mms, where et = ± 1 . There exist inte­

gers d and r such that m = dAs + r and 0 < r ^ As, d ^ 0. Let us consider the s-decomposable number

+

M = p — asj I X ask(m + k f - 2 a sjl ( d + l ) A s+ i y ] ,

k = 1

where j = Ts + 1 — r. The number in the curly brackets differs ,from

A s

N = Y, ask(m + k f by the sign of the term [_(d+ 1);4S+ l ] s.

k= 1

For s ^ 2 we have 2S\AS, hence N = 0(mod As) and M = p + 2 (mod As), because a^ = 1. Therefore M is a s-decomposable number of the form p + 2 + t A s. In virtue of Lemma 4 the number p + 2 is s-decomposable. Thus we have proved by induction that all non-negative integers are s-decom­

posable. Changing all signs in the representation of the number k, we see that — к is also s-decomposable. In virtue of Lemma 3 the theorem is proved.

References

[1] L. E. D ic k so n , Introduction to the theory o f numbers, Dover Pubis., New York 1957.

[2] P. ErdOs, J. S u râ n y i, Vâlogatott fejezetek a szâmelméletb'ôl, TankOnyvkiadô Vâllalat, Budapest 1960.

[3] W. S ie rp in s k i, 250 problems in elementary number theory, American Elsevier Publ. Co., New York 1970.