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C O L L O Q U I U M M A T H E M A T I C U M

VOL. LXVIII 1995 FASC. 1

ON INTEGERS NOT OF THE FORM n − ϕ(n)

BY

J. B R O W K I N

AND

A. S C H I N Z E L (WARSZAWA)

W. Sierpi´ nski asked in 1959 (see [4], pp. 200–201, cf. [2]) whether there exist infinitely many positive integers not of the form n − ϕ(n), where ϕ is the Euler function. We answer this question in the affirmative by proving

Theorem. None of the numbers 2 k · 509203 (k = 1, 2, . . .) is of the form n − ϕ(n).

Lemma 1. The number 1018406 is not of the form n − ϕ(n).

P r o o f. Suppose that

(1) 1018406 = n − ϕ(n)

and let

(2) n =

j

Y

i=1

q α i

i

(q 1 < q 2 < . . . < q j primes).

If for any i ≤ j we have α i > 1 it follows that q i | 2 · 509203, and since 509203 is a prime, either q i = 2 or q i = 509203. In the former case n − ϕ(n) ≡ 0 6≡

1018406 (mod 4), in the latter case n − ϕ(n) > 1018406, hence

(3) α i = 1 (1 ≤ i ≤ j).

Since n > 2 we have ϕ(n) ≡ 0 (mod 2), hence n ≡ 0 (mod 2). However, n/2 cannot be a prime since 1018405 is composite. Hence ϕ(n) ≡ 0 (mod 4) and n ≡ 2 (mod 4). Moreover, n ≡ 1 (mod 3) would imply ϕ(n) ≡ n − 1018406 ≡ 2 (mod 3), which is impossible, since

ϕ(n) ≡  0 (mod 3) if at least one q i ≡ 1 (mod 3), 1 (mod 3) otherwise.

Hence n ≡ 2 (mod 12) or n ≡ 6 (mod 12) and

(4) n − ϕ(n) > 1

2 n.

1991 Mathematics Subject Classification: Primary 11A25.

[55]

(2)

56 J. B R O W K I N AND A. S C H I N Z E L

Let p i denote the ith prime and consider first the case n = 12k + 2. We have q 1 = 2, q i ≥ p i+1 (i ≥ 2). Since

(5)

7

Y

i=2

p i+1 > 1018406,

it follows from (1)–(4) that j ≤ 6 and 1

2

6

Y

i=2

 1 − 1

p i+1



≤ ϕ(n)

n ≤  2/5 if n ≡ 0 (mod 5), 1/2 otherwise.

Hence if n = 12k + 2 satisfies (1) we have either 116381 < k < 141446 or 141446 ≤ k < 169735 and k 6≡ 4 (mod 5).

Consider now n = 12k + 6. Here q 1 = 2, q 2 = 3, q i ≥ p i . By (1)–(5), j ≤ 7 and

7

Y

i=1

 1 − 1

p i



≤ ϕ(n) n ≤ 1

3 . Hence if n = 12k + 6 satisfies (1) we have

103561 < k < 127301.

The computation performed on the computer SUN/SPARC of the Insti- tute of Applied Mathematics and Mechanics of the University of Warsaw using the program GP/PARI has shown that no n corresponding to k in the indicated ranges satisfies (1).

Lemma 2. All the numbers 2 k · 509203 − 1 (k = 1, 2, . . .) are composite.

P r o o f. We have

509203 ≡ 2 a

i

(mod q i ),

where hq i , a i i is given by h3, 0i, h5, 3i, h7, 1i, h13, 5i, h17, 1i and h241, 21i for i = 1, 2, . . . , 6, respectively. Now, 2 belongs mod q i to the exponent m i , where m i = 2, 4, 3, 12, 8 and 24 for i = 1, 2, . . . , 6, respectively.

It is easy to verify that every integer n satisfies one of the congruences n ≡ −a i (mod m i ) (1 ≤ i ≤ 6).

If k ≡ −a j (mod m j ) we have

2 k · 509203 ≡ 2 a

j

−a

j

≡ 1 (mod q j ),

and since 2 k · 509203 − 1 > q j the number 2 k · 509203 − 1 is composite.

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ON INTEGERS NOT OF THE FORM

n−ϕ(n) 57

R e m a r k 1. Lemma 2 was proved by H. Riesel, already in 1956 (see [3], Anhang).

The following problem, implicit in [1], suggests itself.

Problem 1. What is the least positive integer n such that all integers 2 k n − 1 (k = 1, 2, . . .) are composite?

P r o o f o f t h e t h e o r e m. We shall prove that n − ϕ(n) 6= 2 k · 509203 by induction on k. For k = 1 this holds by virtue of Lemma 1. Assume that this holds with k replaced by k − 1 (k ≥ 2) and that

(6) n − ϕ(n) = 2 k · 509203.

If ϕ(n) ≡ 0 (mod 4) we have n ≡ 0 (mod 4) and n

2 − ϕ  n 2



= 2 k−1 · 509203,

contrary to the inductive assumption. Thus ϕ(n) ≡ 2 (mod 4) and n = 2p α , where p is an odd prime. From (6) we obtain

p α−1 (p + 1) = 2 k · 509203.

By Lemma 2, α = 1 is impossible. If α > 1 we have p | 2 k · 509203,

and since 509203 is a prime, p = 509203, α = 2 and 509204 = 2 k ,

which is impossible. The inductive proof is complete.

R e m a r k 2. D. H. Lehmer on the request of one of us has kindly com- puted the table of all numbers not of the form n − ϕ(n) up to 50 000. This table and its prolongation up to 100 000 seems to indicate that the numbers not of the form n − ϕ(n) have a positive density, about 1/10.

This suggests

Problem 2. Have the integers not of the form n − ϕ(n) a positive lower density?

Added in proof ((November 1994). A computation performed by A. Odlyzko has shown that there are 561 850 positive integers less than 5 000 000 not of the form n − ϕ(n).

REFERENCES

[1] A. A i g n e r, Folgen der Art ar

n

+ b, welche nur teilbare Zahlen liefern, Math. Nachr.

23 (1961), 259–264.

[2] P. E r d ˝ o s, ¨ Uber die Zahlen der Form σ(n) − n und n − ϕ(n), Elem. Math. 28 (1973),

83–86.

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58 J. B R O W K I N AND A. S C H I N Z E L

[3] W. K e l l e r, Woher kommen die gr¨ oßten derzeit bekannten Primzahlen? , Mitt. Math.

Ges. Hamburg 12 (1991), 211–229.

[4] W. S i e r p i ´ n s k i, Number Theory , Part II, PWN, Warszawa, 1959 (in Polish).

INSTITUTE OF MATHEMATICS MATHEMATICAL INSTITUTE

UNIVERSITY OF WARSAW POLISH ACADEMY OF SCIENCES

BANACHA 2 ´ SNIADECKICH 8

02-097 WARSZAWA, POLAND P.O. BOX 137

E-mail: BRO@PLEARN.BITNET 00-950 WARSZAWA, POLAND

E-mail: SCHINZEL@IMPAN.IMPAN.GOV.PL

Re¸ cu par la R´ edaction le 11.4.1994

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