PROOF OF D. J. NEWMAN'S COPRIMEMAPPING CONJECTURE

CARL POMERANCE AND J. L. SELFRIDGE

§1. Introduction. In this paper we prove

THEOREM 1. If N is a natural number and I is an interval of N consecutiveintegers, then there is a 1-1 correspondence f: {1, 2,. . . , N} -* I such that (i,f(i)) = 1for 1 s£ i «: N.

We call the function / described in the theorem a coprime mapping. Theorem 1settles in the affirmative a conjecture of D. J. Newman. The special case when/ = {JV + l.JV + 2, ...,2Af} was proved by D. E. Daykin and M. J. Baines [2]. V.Chvatal [1] established Newman's conjecture for each JV ^ 1002. We proveTheorem 1 constructively by giving an algorithm for the construction of a coprimemapping / . This algorithm will be discussed in §2.

If u is a real number and n is a natural number, let D{u, ri) denote the number ofodd integers /, 1 < / < 2n — 1, with <p(l)/l ^ u, where 4> denotes Euler's function. Ifalso k is a natural number, let E(k, ri) denote the maximal number of integerscoprime to k that can be found in every set of n consecutive integers. Thus, forexample, £(3,4) = 2, since in every set of 4 consecutive integers there are at least 2integers coprime to 3 and the set {0,1,2, 3} has exactly 2 integers coprime to 3. Ifn > 1, let p^ri) denote the largest prime not exceeding 2n — \. We shall prove

THEOREM 2. / / n is a natural number and k is odd with 1 < k < 2n — 1 andk + pi(«), then D{<j>(k)/k, n) < E(k, n).

In §2 we shall show that Theorem 1 is a fairly simple corollary of Theorem 2. Theremainder of the paper then will take up the proof of Theorem 2.

Note that the function D(u, ri) is related to the well-known distribution functionfor (j>:

D^u) = lim - £ 1 .

In fact, if we let D(u) = \imn^ ^Diu, ri)/n, then it can be seen that, like D^u), D(u)exists for all u, D(u) is strictly monotone on [0, 1], D{u) is continuous, and D(u) issingular on [0,1]. We have £>0(«) = (D(u)+D(2u))/2.

For k fixed it is easy to see that E(k, ri) ~ (cj)(k)/k)n as n -+ oo. We thus have thefollowing consequence of Theorem 2.

COROLLARY. For every u ^ 0, D(u) < u.

[MATHEMATIKA, 27 (1980), 69-83]

70 C. POMERANCE AND J. L. SELFRIDGE

We do not use computers in the proof, although many calculations wereperformed out of convenience on a hand calculator. However, the proof relies quiteheavily on the computer-assisted results of Rosser and Schoenfeld [4].

We take this opportunity to thank Paul Erdds for calling our attention toNewman's conjecture. We should also remark that a key step in our proof—namely,the use of the sets J,(P) in §7—is suggested by an old argument of Erd6s (Theorem 3in [3]). We also wish to thank M. Mendes France for informing us of [1].

§2. The algorithm. In this section we shall inductively describe an algorithm forthe construction of a coprime mapping/. The algorithm will make use of Theorem 2,which shall be assumed as true in this section. Thus this section will show thatTheorem 2 implies Theorem 1.

If JV = 1, then there is only one mapping/: {1} -> / , and certainly/is a coprimemapping. If N = 2, then let / ( I ) be the even member of / and let /(2) be the oddmember of / . This / i s a coprime mapping.

Say now N ^ 3 and we have given algorithms for the construction of coprimemappings for all M < N. Let / be a set of N consecutive integers. We first describe /at the odd members of {1,2,... , AT} and then use our induction hypothesis todescribe / at even arguments.

Let n = N — [JV/2] denote the number of odd numbers in {1, 2,. . . , N}. Thenn ^ 2. Label these odd numbers kl,k2,...,kn where (/>(&()/&; ^ <j)(ki+1)/ki + l for1 ^ i < n. Note that kn^1 = px(n) and kn = 1. We first describe how to choose/(*!>, ...,/(*,,-2).

Now / has either n o r n - 1 even numbers. If they are each divided by 2, then weobtain a string of consecutive integers. Thus for each i, there are at least E(kh n — 1)even integers in / that are coprime to kt.

By Theorem 2, if n Ss 3,

E(k,, n-1) ^ E(kt,n)-\> D^)/^, n) ^ 1,

so that there are even numbers in / coprime to k1. Let /(fcj be the least such number.Say i < n —2 and f(kx), ...,/(&,_!) have already been defined. By Theorem 2,

£(/c;, n - 1 ) > E(kh n)-\> />(#*,)/*„ n) > i,

so that there are at least i even numbers in / coprime to fe;. Thus we may let /(fc,) bethe least even number in / coprime to kt and not equal to f{kx\ ...,/(&;_!).

We now define f{kn_i). If / has n even numbers, there are 2 of them left whichhave not yet been used as a value of/. Moreover, kn_x = p^ri) does not divide bothof them since half of their difference is at most n — 1, while by Bertrand's Postulate,p^ri) > n. Thus we let f(kn_l) be the least of the remaining even numbers of/ whichis not divisible by Pi(n). If/ has only n — 1 even numbers, then both endpoints of/ areodd and not both are divisible by p^n) (half of their difference is n —1). So we let/(/cn_!) be the least endpoint of / which is not divisible by Pi(n).

In either case, we have exactly one even number left in / which has not yet beenused as a value of/. We let f(kn) be this number.

Now we describe how to define / at even arguments, using essentially the sameidea as Theorem 2 in [1]. Let J be the set of remaining members of / not used

PROOF OF D. J. NEWMAN'S COPRIME MAPPING CONJECTURE 71

already as values of/. Then J is an interval of [N/2] consecutive odd numbers. Letm equal the product of the odd primes not exceeding [N/2]. Let/' = {(j + m)/2: j e J}. Then / ' is an interval of [N/2] consecutive integers. By ourinduction hypothesis there is a construction for a coprime mappingg: {1,2,..., [N/2]} -+ / ' . If 1 < fc < [N/2], let f(2k) = 2g(k)-m. We have thusdescribed a 1-1 correspondence from {2,4, ...,2[N/2]} to J and for each k,1 < k < [N/2],

(2k,f(2k)) = (2k,2g(k)-m) = l.

This completes the construction of the coprime mapping / .Before we proceed to the proof of Theorem 2, we describe a simpler algorithm for

the construction of a coprime mapping. We do not have a».proof that this simpleralgorithm is always successful, but we feel that a proof probably could be providedusing the methods of this paper. As in the above algorithm, we may assume N ^ 3.Relabel {1, . . . , N} as blf..., bN where 0(&,-)/fej < (f>(bi+i)/bi+1 for 1 < j < JV. If it isnot the case that both of /'s endpoints are odd, inductively define /(b,) as the leastnumber in / coprime to bt and not equal to /(b^, ...,/(fo;_ j). If both of /'s endpointsare odd, first define f{bN_t) as one of the endpoints, and then inductively define f(bt)for i ± N — 1 as the least number in / — {f{bN_l)} coprime to bt and not equal tof(bi), ...,/(£;_!). The need for the special treatment of bN_x in the latter case can beseen from the example N = 9,1 — { —1,0,..., 7}.

§3. Preliminaries. If k is a natural number, let co(k) denote the number of distinctprime factors of k.

LEMMA 1. lfk>\, then

£ ( f c n ) > n

Proof. If k0 is the largest square-free divisor of k, then E(ko,n) = E(k,ri),^oV^o = 4>(k)lk, and co(/c0) = w(k). Thus we may assume k itself is square-free. If mis any integer and if j > 1 is an integer, then

< 1.

Thus

m < i ^ m + n j\k m < i ^ m + n

>n+ Xi\k

j >

72 C. POMERANCE AND J. L. SELFRIDGE

LEMMA 2. / / 1 < k < 2n— 1 and ifp is the largest prime factor ofk, then,

\ k p -

Proof Write k = p'm where p J m. Thus pm < In, so that

1 — m/n > 1 — 2/p .

ThusE(k,n) ^ £(m,n)-([n/p] + [n/ni] -n/p -

m n

m pj p \ k ' p - 1 p

LEMMA 3. If m > 1 is odd, and n is a positive integer,

1 1

_V

< * < 2 n - l2 1 k , m \ k

1 - ^m 2 2m

Proof. The quantity in the absolute value signs has its minimal value whenIn — 1 = m —2; the value is — l/2 + l/(2m). The maximal value is attained when2 n - l = m; the value is l/2-l/(2m).

LEMMA 4. / / m is odd, let

F(m,n)= 1-1 =S k =S 2 n - l2 / k , (»>.*) > 1

ThenF(m, ri)

F(IO5, n) < M«

Proo/ The maximum value of F(m, ri) — (1 — <t>(m)/m)n for all n can be computedby evaluating this difference for any m consecutive choices for n. This is how weestablished the latter two assertions. To see the first statement, note that by Lemma3,

F(m,n)=; I m.j >

JI m,j

1 < k < 2 n - l

j I m,j

PROOF OF D. J. NEWMAN'S COPRIME MAPPING CONJECTURE 73

< 1 -m

m Jwhere, as in the proof of Lemma 1, we assume m is square-free.

§4. The proof of Theorem 2 for n < 1000. We first note that for any fixed n,Theorem 2 may be numerically checked as follows. Order the odd numbers k,1 ^ k < 2n —1, as ki,k2,...,kn where </>(k,)//c; < cj)(ki + i)/ki+l for 1 < i < «. Thenumbers E(kh n) are each evaluated. We then check if i < E[kh n) for each i < n — 2.If so, Theorem 2 has been established for n. Indeed, if 4>{k)/k = $(/)//, then k and /must have the same set of prime factors, so that E(k, n) = £(/, n). Thus if1 ^ i ^ n — 2 and

:;, n) = j < E(kj, n) = E(kh n).then

We follow the above procedure for n ^ 16. Of course there is nothing to provefor n = 1,2. Moreover, one can easily see that if Theorem 2 is true for n — 1 and2n — 1 is prime, then Theorem 2 is true for n as well. Thus the above procedure needonly be carried out for n = 5, 8, 11, 13, and 14. These calculations are displayed inTable 1.

We now follow a different procedure to prove Theorem 2 for n in the interval17 ^ n < 1000.

PROPOSITION 1. Theorem 2 is true if

(1)

E(kh 5)

Table 1

E(kh 8)

1153

4 55 76 6

6117

•i, 11)

1155

2215

7 8 911 13 1710 10 10

t,, 13)

1156

2 321 36 8

5510

62510

7711

81111

91312

101712

111912

£(*„ 14)

1157

2217

5279

6511

72511

8712

91112

101312

111713

121913

74 C. POMERANCE AND J. L. SELFRIDGE

Proof. Let n ^ 17, k odd, l < k < 2 n —l,fe^= px(n), and assume (1). Thus k isdivisible by no prime p < V(2« — 1), so that k is itself a prime and fe > J(2n — 1).Denote fc by p, where p; is the i-th largest prime not exceeding 2n — 1. Thus

D{4>{k)lk, n) = D{\ - 1/p,., «) = « - i , (2)

for the only odd / ^ In — 1 with (/>(/)// > 1 — l/pt- are p 1 _ 1 , p I _ 2 , . . . , ? ! , 1. Now byLemma 1,

Thus to show D((j)(k)/k, n) < E(k, ri), we need only show that

n/Pl<i-l. (3)

Now (3) is certainly true for n < pt < px. By Rosser and Schoenfeld [3], the largesti0 with n < p,0 < pt is exactly

n(2n)-n(n) > n/(2 log n) for all n > 5-5 .

Thus if ^J(2n — 1) < p, ^ n, we have

n/Pi < n/V(2n-1) < n/(2 log n) < i0 < i - 1

which holds for all n ^ 16. Thus (3) holds for all i > 1.For future reference we note that (2) implies for all n > 1,

(4)

For the remainder of this section we assume 17 < n ^ 1000, k is odd, k ^ 2n — 1,and

Case 1. 0(/c)/A: ^ 120/143 = (10/ll)(12/13). Using (4), a table of primes up to2000, and a hand calculator, we verify that D(4>(k)/k, n) < (0-712)n (cf. Lemma 1 in[1]). If aik) $s 3, then k 5= 13 . 17. 19 > 2 n - l , so we may assume oik) < 2. ByLemma 1, we thus have E(k, ri) > (120/143)n-3. Thus D(<f>{k)lk, n) < E{k, ri) for alln Ss 24. Assume 1 7 ^ n < 24. If oik) ^ 2, then k ^ 11. 13 > 2 n - l , acontradiction. So assume co(k) = 1. By Lemma 1, E(k,n) > (120/143)n —1. ThusD{4>{k)/k, n) < E{k, ri) for 17 ^ n < 24.

Case 2. 120/143 > 4>{k)/k 5* 720/1001 = (6/7X10/11)(12/13). Since 11. 13 . 17 >2000, the condition <t>(k)/k < 120/143 implies k is divisible by 3, 5, or7. Moreover, k is not a power of 7. Thus by Lemma 4, D((j)(k)/k, n) ^ (19/35)n-5/7.Indeed, this is a direct consequence of Lemma 4 if n ^ 25, and if 17 ^ n < 25, wenote that £(105, ri) < (19/35)n + 2/7. Now the only odd k < 2000 with co(k) Ss 4 are

3 . 5 . 7 . 1 1 , 3 . 5 . 7 . 1 3 , 3 . 5 . 7 . 1 7 , 3 . 5 . 7 . 1 9 , (5) j

and none of these satisfy the conditions of Case 2. Thus oj(k) ^ 3. By Lemma 1, IE(k,n) > (720/1001)n-l. Thus D{4>(k)lk,n) < E(k,n) for 36 ^ n < 1000. So!

PROOF OF D. J. NEWMAN'S COPRIME MAPPING CONJECTURE 75

assume 17 < n < 36. Then oik) ^ 2, so that E(k, n) > (720/1001)n-3. ThusD((j)(k)/k, n) < E(k, n) for these n as well.

Case 3. 720/1001 > <j}(k)/k > 48/77 = (4/5)(6/7)(10/ll). Since k < 2000, we seethat 3 | k or 5 | k. Moreover k is not a power of 5. Thus for 17 < n ^ 1000, Lemma 4implies D(<£(/c)//c, n) < (7/15)n—4/3. Since none of the integers in (5) satisfy thecondition of Case 3, we have co{k) < 3. Thus E(k, n) > (48/77)n — 7, so thatD((j)(k)/k,n) < E(k,n) if n ^ 37. But if 17 ^ n < 37, then <o(k) < 2, so that£(fc, n) > (48/77)n-3 and D{4>{k)lk, n) < E(k, n).

Case 4. 48/77 > <p(k)/k. Since 5 . 7 . 11 .13 > 2000, we have 3 | k and k is not apower of 3 so that Lemma 3, implies D{<f>(k)lk, n) ^ (l/3)n —8/3. Since co(k) < 4, wehave <t>{k)/k $s (2/3)(4/5X6/7)( 10/11) = 32/77. Thus -Lemma 1 impliesE(k, n) > (32/77) n - 1 5 and we see that D((j)(k)/k, n) < E(k, ri) if n 3? 150. So assumen < 150. Then oAk) ^ 3 and 0(fc)//c ^ (2/3)(4/5)(6/7) = 16/35. We haveE(k,n) > {16/35)n-7 and D(4>(k)/k, n) < E(k, n) if n ^ 36. So assume 17 < n < 36.Then co{k) < 2, £(fc, n) > (8/15)n-3 (note that (2/3X4/5) = 8/15), and

c, n) < £(/c, n).

§5. The proof of Theorem 2forn> 1000: r/ie range <j)(k)/k ^ l-( logw)-1.

PROPOSITION 2. Theorem 2 is true for

1, l - ( 2 n - l ) - 1 / 2 ] . (6)

Proof. Let n > 1000, 1 < f e ^ 2 n - l , / c odd, and assume (6). We distinguish thetwo cases ca{k) ^ 5, co(k) ^ 6.

If oik) ^ 5, then by Lemma 1,

E(k,n) > (l

But by (4) and Rosser and Schoenfeld [4],

, n)<n + V(2n - 1 ) -2n/(log (2n) - 1 / 2 ) . (7)

Thus D((j)(k)/k, n) < E(k, n).So we now assume a>(k) ^ 6. Then p, the largest prime factor of k, is at least

5 log n + 1 . For if p < 5 log n +1, then

1 + 5 log n A - l + 51ogn/ V 5 log n

6 15 1> 151ogn ' 251og2n logn'

a contradiction. Thus by Lemma 2,

76 C. POMERANCE AND J. L. SELFRIDGE

\ \ l / \l o g n / \ 51ogn/ 5 log n

5 log n„ _ , .

Thus by (7), D(<t>(k)/k, n) < E{k, n).We now note that Proposition 2 in conjunction with Proposition 1 prove

Theorem 2 for all k in the range <t>(k)/k > 1 -(log ri)~x.

§6. The range <f>{k)/k s£ 0-623.

PROPOSITION 3. For all n > 1000,

Z k/4>(k) < (l-3)n .

Proof. Throughout the proof, the letters d,j, k, m, will stand for positive oddintegers. Let h be the multiplicative function such that h(p) = l/(p —1) for primes pand h(pl) = 0 for i ^ 2. Then n/(/>(n) = £ /i(/). Thus

(c<2nz z

fc < 2n d | fc d <2n m < Inji

d < In d <2n

Now

d <2n P>2

(8)

1-2958 , (9)

where C is Riemann's function.Also let H denote the multiplicative function with H(p) = p'1^ — I)"1 ,

i/(p2) = -ff(p), and H(p') = 0 for i > 3. Then h(ri) = £ H(/)//n. Thus

d <2n j\dZ H(j) Z< 2n m < 2n/j

2(2 fr2irf3i\^ • H ^ 2 < 1-6791+(0-8396) log (2«).3 4(6) /

(10)

PROOF OF D. J. NEWMAN'S COPRIME MAPPING CONJECTURE 77

From (8), (9), (10), we have

X k/<Mk) < (l-2958)n-0-6478 + 0-8396 + (0-4198)log(2n)

PROPOSITION 4. For all n > 1000 and 0 < u < 1, D(u, n) < (0-3)wn/(l -u).

Proof. Throughout the proof, the letter k will stand for a positive odd integerparameter. Let n > 1000, 0 < u < 1 be fixed. Let c be such that D(u, n) = en. Then,using Proposition 3,

en = X 1 < u X fc/#*) = « I */#*)-« Zk < 2n k < 2n k < In k <7n

<Hk)/k < u 0(fc)/t < u *(li)//k > u

<(l-3)wn-u X 1 = (l-3)wn-M(l-c)n,

so that c < (0-3)M+CM and our conclusion follows.

PROPOSITION 5. Theorem 2 is true if 4>(k)/k < 0-623.

Proof Let n > 1000, 1 < k < 2n-l, k odd, </>(£)/£ < 0-623. We distinguishtwo cases: a>(k) < 7, a>(/c) > 8.

Say co(/c) < 7. By Lemma 1, E(k, n) > ((/>(k)/k)n-127. Thus using Proposition 4,D(4>{k)lk, n) < E(k, n) if (with u =

un-127 > (0-3)un/(l-u);

that is,n > 127(1-M)/M(0-7-M). (11)

Now the maximal value of u is 0-623 and the minimal value is

n o-i/p).3 « p « 19

But the maximal value of the right side of (11) for the stated range of u is below 1000,so (11) holds.

Now say co(k) > 8. Let p denote the largest prime factor of k and write k = p'm,p\m. Then p ^ 23. Let u = 4>(k)/k. We have

p'1 =(p-iy1um/<fi(m)^(p-ir1u n «/(«-!)2 <q <p

<(uP.2) [I « / ( « - 1 ) < ( 0 - 1 3 2 9 ) M , (12)2 < a < 23

78 C. POMERANCE AND J. L. SELFRIDGE

where we use the fact that the function

(p-ir1 n «/(«-!)2<q <p

defined for odd prime arguments p is decreasing. From (12) and Lemma 2 we have

E(k,ri) > ( 2 1 / 2 2 - 0 - 1 3 2 9 ) H H - 1 > (0-8216)«n-l.

Thus by Proposition 4, D(<f>(k)/k, n) < E(k, n) if

(0-8216)un-l

that is,

(13)uV0-5216-(0-8216)u

Now the maximal value for 0 < u s£ 0623 of the right factor in (13) is less than 40.Thus the proof will be complete if we can show n > 40/u.

From Rosser and Schoenfeld [4]

\ju = k/<f>(k) < (1/2)^ loglog k + 1-26/loglog k for k ^ 3,2 \ k.

Thus it only remains to note that for all n > 1000,

n > 40((l/2)e7 loglog (2n)+ 1-26/loglog (2n)) > 40/u .

§7. 77M; rangre 0-623 < (j){k)/k < 1 -(log n)'1. If P is a prime, P ^ 7, let S(P, n)denote the set of odd k, l ^ / c ^ 2 n — 1, with /c divisible by at least t + 1 distinctprimes in some set defined by

( [0, P), if t = 0 ,Jr = J,(P) =

U S ^ ^ ^ ' P ) , if t>l.

If fe is odd, 1 «S k ^ 2 M - 1, and /c ^ S(P, n), then

where a, = a,(P) is defined to be the product of the first t primes in J, (if J, does nothave t primes, then a, is the product of all the primes in Jt). Write

r = l

Thenn) ^ \S(P, n)

PROOF OF D. J. NEWMAN'S COPRIME MAPPING CONJECTURE 79

Let D(P, t, n) denote the number of odd fc,l<fc<2« — 1, such that k is divisible byit least t +1 distinct primes in Jt. Thus

D(u(p),n)^ £ D(P,t,n). (14)1 = 0

We now proceed to define the column headings in Table 2.

Definition ofv(P). We define v(P) as a positive quantity satisfying the inequality>(P) < u(P) according to the following scheme. Let

= 0-704

^orP ^ 11, let

2-25

-1 p ^ p y PU

Definition of w(P). We shall define w(P) as a positive quantity, such that if' > 7, then w(P) ^ v(P') where P' is the prime just before P, according to theallowing scheme.

w(7) = 0-623 .•or 11 sS P < 37, let

ly Rosser and Schoenfeld [4], if P $s 41, then P' > 7P/8. So we may take

Definitions ofd(P) and c(P). We shall define positive quantities d(P), c(P) so that>r all n ^ 1,

), n) ^ D(u(P), n) < d{P)n+c(P). (15)We let

d{l) = 0-543 , c(7) = 6 .

remains to show that (15) holds for P = 7. First we note that Lemma 4 implies

0(7,0,1) < (7/15)n + 2/3.

ince D(7,1, n) is the number of odd k, l < / c ^ 2 n — 1, divisible by two distinctrimes in {7, 11, 13, 17, 19}, Lemma 3 implies

D(l, l ,n) < (0-069)n + 5 .

say, a result that is valid for all T > 1. So '

\t*i ' iS

f) p) +2n

( l 1/pY\peJl J

> -1 - a(63)

(0007)n.

Thus by (14), D(M(7), «) < (0-543)n + 6.We let

= 0-625, c(ll) = 12.

To show (15) holds, we note that Lemmas 4 and 3 imply

D(U, 0, n) ^ (19/35)n+9/7 , D(ll, 1,«) < (0064)n +10-5 .

Furthermore, by (16),

£ D(U,t,n)<2n j1=2 1-2 \i>ej,

< 2n(e*(")-l-a(33)-£a(33)2) < (0018)n .

Thus D(M(11), n) < (0*625)71 +12.For 13 «£ P ^ 37, we let (cf. (16))

2 < p < P

c(P) - 2"p>-3;

and for P > 41, we let

•* - a(3P)

Where the last inequality is valid for all P ^ 5 from the estimates of Rosser andSchoenfeld [4]. Also by (16),

X D(P,t,n)<2n £ ( £ 1/pY '/(I+1)!1=1 1=1 \n<=y, /

\2

[ 1/p +2H £ «(;= J| / 1 = 2

P < 41

2» £ «(.

« ( I 1/PVfJi

P < 4 1

P ^ 41

Thus (14) implies (15).

Definition of pr (P). For 7 < P < 37, we define pr(P) as the least prime p sothat there is a k with 6 distinct prime factors, the largest of which is p, and<t>{k)jk ^ w(P). For P Js 41, we let

pr(P) = 35P/18 = 5(1 -

We now note that if k has at least 6 distinct prime factors and 4>(k)/k > w(P), thenone of these primes is at least as big as pr (P). For if not,

<Mk)/k < (1 - 1/pr (P))6 = (1 -18/(35 P))6

< 1 -6(18/(35P))+ 15(18/(35 P)f

< 1-5(18/(35 P)) = w(P).

Definition of x(P). We shall define x(P) so that if k is divisible by at least 6distinct primes and <f>{k)jk 3* vv(P), then

E(k,n) > x(P)n~\ . (17)

82 C. POMERANCE AND J. L. SELFRIDGE

For 7 < P < 37, we let

x(P) = w(P)[l - l/(pr (P) -1 ) j - 1/pr (P),

so that (17) follows from Lemma 2. For P ^ 41, we let

x(P) = 1 -3-6/P < (1 - 18/(7P))(1 -18/(35 P))-18/(35 P)

< w(P)(l - l/(pr (P) - 1 ) ) - 1/pr (P),

so again (17) follows from Lemma 2.

PROPOSITION 6. / / /c is odd, 1 < fc < 2n - 1 , w(P) ^ <t>{k)/k ^ D(P), and

< x(P), then D{4>(k)lk, n) < E(k, ri)for all n > M^P), with

Proof. From (15), we have

D{<p(k)/k,n) < d(P)n + c{P).

Say co(/c) < 5. By Lemma 1,

E(k,n) > {4>{k)lk)n-31 S* w(P)n-31 .

The assumption d(P) < x(P) implies d(P) < w(P). Thus D(<p(k)/k, n) < E(k, n) for an > (c(P) + 3l)/(w(P)-d(P)).

Now assume co(k) ^ 6. By (17), D((j>{k)/k, n) < E(k, ri) for a( ) / ( )

PROPOSITION 7. Theorem 2 is true if 0-623 < (j>(k)/k < 1 —(log n)"1 .

Proo/ Let n > 1000, fe odd, 1 < k ^ 2 n - l , 0-623 < ^(k)/fe < l -( logn)~There exists a prime P ^ 7 such that w(P) ^ (j)(k)/k < t)(P). Thiw(P) < 1 -( log n )~ \ so that

n > max {1000, d1-*^} = M2(P), (I1

say. By an examination of Table 2, we see that the condition (19) implies tlcondition (18) for all P 5= 7. Thus by Proposition 6, D(4>{k)lk, n) < E(k, n).

In comparing the last two columns of Table 2 for P > 41, it is helpful to noithat M2(P) = e1Pin and since <x(P) < 1-316/logP,

M^P) < 6(logP)(2p/(logi>-15)

PROOF OF D. J. NEWMAN'S COPRIME MAPPING CONJECTURE

Table 2

83

p

11113171923293137

<AP)

•704•795•826•867•881•902•922•927•939(a)

w(P)

•623•704•795•826•867•881•902•922•927(b)

d(P)

•543•625•706•729•727•742•761•759•111

(c)

c(P)

6128163264128256512

(d)

pr(P)

293141435961738997(e)

x(P)

•566•648•750•783•835•849•875•900•907

(f)

Mt(P)

463566439485450684113218233947

M2(P)

10001000100010001842446227013369724889691

(a) 1-2-25/P

(b) 1 -

(c) 1 -2e-y iog P -2e- ' / log 3 P + 2(ea(p)-1 -oc(P))

where a(P) = log (l +(log 3)/log P) + I/log 2 (3P) +1/(2 log 2P)

(d) 2p/(Iogi>~15)~3

(e) 35P/18

(f) 1-3-6/P

Round-off notes

bounded down:

bounded up:

v(P), w(P), x{P), M2(P)

d(P), Mt(P)

The entries in the last two columns (see (18), (19) for definitions) are computed from

he rounded numbers appearing in the other columns of the table.

References

. V. Chvatal. "A remark on Newman's conjecture", Proc. Washington State Univ. Conf. on NumberTheory, 1971, 113-129.

I D. E. Daykin and M. J. Baines. "Coprime mappings between sets of consecutive integers",Mathematika, 10 (1963), 132-136.

I. P. Erd6s. "Some remarks about additive and multiplicative functions", Bull. Amer. Math. Soc, 52(1946), 527-537.

L J. B. Rosser and L. Schoenfeld. "Approximate formulas for some functions of prime numbers", IllinoisJ. Math., 6 (1962), 64-94.

Jniversity of Illinois at Urbana-Champaign,Jrbana, Illinois 61801ndJniversity of Georgia,Uhens, Georgia 30602.

Mathematical Reviews,Lnn Arbor, Michigan 48109ndNorthern Illinois University,)eKalb, Illinois 60115.

10A99: NUMBER THEORY; Elementary numbertheory.

Received on the 3rd of July, 1979.