On Relatively Prime Odd Amicable Numbers
By Peter Hagis, Jr.
Abstract. Whether or not a relatively prime pair of amicable numbers exists is still an open
question. In this paper some necessary conditions for m and re to be a pair of odd relatively
prime amicable numbers are proved. In particular, lower bounds for m, n, mn and the
number of prime divisors of mn are established. The arguments are based on an extensive
case study carried out on the CDC 6400 at the Temple University Computing Center.
1. Introduction.
Two positive integers m and n sire said to be amicable if
(1)
o(m) = m 4- n = o(n)
where o-(fc)is the sum of the positive divisors of k. To date almost 900 pairs of
amicable numbers have been found (see [1], [2], [3], [4], [7] and [8]), none of which is
relatively prime. In [6] Kanold has shown that if m and n are relatively prime
amicable numbers then mn > 4 • 1046.In [5] the present author has shown that if m
and n are relatively prime amicable numbers of opposite parity then mn > 1074.
The purpose of the present paper is to improve Kanold's lower bound for mn in case
m and n are relatively prime odd amicable numbers. To be precise, we shall prove
the following
Theorem. If
U
(2)
N
m = II nbi,
¿=i
n = Il s/5' (whereM + N = T)
y=i
are amicable numbers such that the odd primes r, and Sj are distinct then
(a) if (mn, 15)
(b) if (mn, 15)
(c) if (mn, 15)
(d) if (mn, 15)
=
=
=
=
1, then T = 606 and mn > 10I9U;
5, then T = 140 and mn > 4-10354;
3, then T = 53 and mn > 3-10118;
15, then T ^ 21 and mn > 10".
The proof involves an exhaustive, and rather exhausting, "case" study. This
was carried out with the aid of the CDC 6400 at the Temple University Computing
Center.
2. Some Preliminary
notation pa||fc means that
a pair of relatively prime
plicative property of oik)
Results. In what follows p and q denote primes and the
pa\k but pa+1\k. m and n will always be understood to be
odd amicable numbers. From (1) and (2) and the multiwe see that
(3)
m + n=
where, for convenience,
Lemma
Uo-irb) = U*ise)
we have omitted the subscripts.
1. If q\mn and pa\\mn,
then q\o(pa).
Proof. If q\mn and </|a-(p°), then from (3) we see immediately
This is impossible since (m, n) = 1.
Received August 6, 1968, revised October 10, 1968.
539
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
that q\m and q\n.
540
PETER
HAGIS, JR.
Lemma 2. If q\mn, pa\\mn, and p = qk — 1 then 2\a.
Proof. If p = qk — 1 and a is odd, then o(p") = 1 4- p 4- p2 4- ■■■ -\- pa =
1 — 14-1—
••• — 1 = 0 (mod q). Since this contradicts Lemma 1 we conclude
that a is even.
Now, since m y^ n we have (m 4- n)2/mn > 4. Therefore, letting JjVb • JJsc =
pi01 -p2a2 ■ ■ ■pTaT where
p,: < p¡ if i < j, we have from
Lemma 3. If mn = H¿=i p,ai, then 4 < ULi
(3)
<x(p¿ai)/p¿oi-
3. The case (inn, 15) = 1. In the sequel we shall denote the ,/th odd prime by
Pj. If (mn, 15) = 1 then px = 7 = Pd. Since o(pa)/pa = (p - p~a)/(p - 1) <
p/(p — 1) we see from Lemma 3 that
t
r+2
i< Upi/iPi1) = IlTVÍPy-l).
¿=i
y-i
It follows that if N is the smallest integer such that XI)=3 Pj/iPj — 1) > 4, then
T ^ N - 2 and mn = JI^-s 7Jy.Making use of the CDC 6400 it was found that
N = 608 and mn = 7-11 -13 • • • 4483 > 1019u.
4. The case (mn, 15) = 5. In this case pi = 5 = P2. Therefore, if N is the
smallest integer such that HyLj Pj/iPj — 1) > 4, then T = N — 1. It was found
that N = 141. From Lemma 2 we see that if p"||mn and p = 5k — 1 then 2\a. We
conclude that mn ^ 5-192-292-7T(7,1093) > 4-10364where 7X7,1093) denotes the
product of the 137 primes between 7 and 1093, inclusive, which are not congruent
to —1 modulo 5.
5. The case (nn, 15) = 3. In this case pi = 3 and p2 = 7. It follows that
-i < n¿-1 v.nvi -1) = 1-53n= 3 PYiPj -1)
so that if N is the smallest integer such that Ipy=3 Pj/iPj
~ Y > 8/3, then T =
N — 1. It was found that N = 54. From Lemma 2 we see that if pa\\mn and p =
3/v - 1 then2|a.
We conclude that mn = 3-ll2-172-232-(7(7,571)
> 3-10118. Here
G(7,571) is the product of the 49 primes between 7 and 571, inclusive, which are
congruent to 1 modulo 3.
6. The case (mn, 15) = 15. This case is far more troublesome and requires the
examination of a multiplicity of subcases before the lower bound given in (d) of
our theorem can be established. We shall present the results of our investigation in
tabular form after some preliminary remarks.
We assume that 3"||mw, a = 1; o^jran, b — 1; 7c\\mn, c i£ 0; lld||mn, d 5ï 0;
13f|[mn, e = 0; 17/||mw, / ^ 0; 19"||ran, g ^ 0; 23;i|mn, h ^ 0; 29r||ran, r ^ 0;
31s]|mn, s Sï 0. Since <r(33) = 40 and 5|mn we see from Lemma
Lemma 2, since 15 \mn, b = d=f=g
and o-(73) = 400, from Lemma
1 that a ^ 3. From
= h = r = 0 (mod 2). Since c(72) = 57
1 we have c = 0, c = 1, or c = 4. Since cr(192) =
3-127, g = Oorg = 4.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
ON RELATIVELY
12
A
o o1—1
O
I© CO
A A
A A
co A
A
CO
oo
m
A
01
A CO
o
iX
A
S
CO
CM
*rt
=
**i I
CO
Os
—
01
CO
Ol
I-
Ft—1
\h t-.
r~
Ss
Ss
CO
CO
*rH
1—I
•H"
CO
1—I
t—1
(O
1-
co
io
CO
1—I
Si
1>
co
CO
—
—
-H
—
CO
5s
CO
01
M<
CO
3s
Oh
IUs
CO
c
<
i
Oh
i^
Oh
CM
10
co
Ol
—
I« CO A
A A oo
CO
Ol
US
co_
CO
Ol
iA A
2
A A
Ci
| oo
A
541
PRIME ODD AMICABLE NUMBERS
i>
US
io
I-
CO
co
CO
US
to
o
CO
to
■o
CO
1^
t£>
US
IO
CO
CO
CO
01
oq
CO
eq
—
CO
05
CO
"9
00
or;
C-1 oi
CO
01
o
o
«s
«s"
»o
CO
us
co
CO
o
o
CO
rV
—I
rH
eq
o
o
o
"+1
b"O
CO
01
US
hi
PC
<
o
A
!-s
A
I©
©
II
■»
All
¡•a
A
o
co
©"
AllI
A
o
A
All
A
CO
All
co
All
e
©
o
o
A
All
e
cT
II
%
cT
A
»
co
AI!
S3
©
II
Os
©"
A
©
All
c
5rs
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
All
CO
CO
All
AIM
ö
oo"
co"
II
All
ö
co"
All
a
cT
A
A
AIM
o
A
o~
o
A
AI
it
542
PETER
HAGIS, JR.
Now assume temporarily that 7]win. Since c(35) = 4 •7 • 13, a ^ 5. Since o-(l I2) =
7-19anda-(ll4) = 16105,d = 0ord = 6. Since <r(13)= 14, o-(132)= 3-61,<r(133)=
2380, e = 0 or e è 4. Since <r(232)= 7 -79, h = 0 or A = 4.
If ll|mn then a^4,
since ct(34) = 121; and 6^4,
13|mn then r = 0 or r ^ 4, since <r(292) = 13-67.
From Lemma 3 it follows that
(4)
since <x(54)= 11-71. If
i < ABCt-H
H pi/ipi - 1) g ABCi=H
II* 7V(P; - 1) •
Here Tí = 3 if 7\mn, 77 = 4 if 7|win. The asterisk indicates the (possible) omission
of certain factors due to restrictions on mn. For example, if 17 twin then 17/16 is
omitted. If 52||win then, since <r(52) = 31, 31 twin and therefore 31/30 will be
omitted. The number of factors in YL* is T — H -\- 1. A = o-(3°)/3° if the value
of a is specified ; A = 3/2 otherwise. B = o-(56)/5i,if 6 is specified; B = 5/4 otherwise.
C = g(Jc)/7c if c is specified; C = 7/6 otherwise.
From (4) we see that lower bounds for T, and consequently
mn, can be de-
termined by finding the smallest integer N such that ABC Tlfln Pj/iPj — 1) > 4.
Our results appear in the accompanying table.
In each case PY(p, q) denotes the product of the k primes between p and q,
inclusive, which are not congruent to —1 modulo any prime known to be a divisor
of win. For example, in the last case
Pu(31,397) = 73-127-157-163-193-211-277-283-313-331-397.
None of these primes is congruent to — 1 modulo p where 3 = p á 31.
In this case, since <r(313) = 2-157, we can improve the lower bound
to 36567411613417219423429431
-73• 127• 157• 163• 193-211 -277-283-331 -397-421 >
1067.
Since the cases discussed are mutually
exclusive and exhaustive
we conclude
that T = 21 and mn > 1067if 15|mn.
7. Lower bounds for m and n. Without loss of generality we assume that
m < n. If n < 4m then from our theorem we have 4wi2 > mn > 1067.It follows that
m > 1033.
If n > 4m then from (1) we have o(m)/m
=14n/m > 5. Arguing as in Section
3 we see that if N is the smallest integer such that YLf-i Pj/iPj — 1) > 5 then m
has at least N different prime divisors and m — Hf=i P'¡. It was found that N = 54
and m = 3-5 • • • 257 > 8-10102.
We have proved the following
Corollary.
If m and n are relatively prime odd, amicable numbers then m > 1033
and n > 1033.
This improves Kanold's result [6] that n > m > 1023.We also remark that the
adjective "odd" can be omitted in the statement of the corollary. For in [5] it has
been proved that if m and n are relatively prime amicable numbers of opposite
parity then m > 1036and n > 1036.
Acknowledgements.
The author
would like to thank
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
the referee for several
ON RELATIVELY
543
PRIME ODD AMICABLE NUMBERS
helpful suggestions. He would also like to express his appreciation to Mr. John
Bieler of the Temple University Computing Center for his cheerful cooperation.
Department of Mathematics
Temple University
Philadelphia, Pennsylvania 19122
1. J. Alanen,
O. Ore
& J. Stemple,
"Systematic
computations
on amicable
numbers,"
Math. Comp., v. 21, 1967, pp. 242-245. MR 36 #5058.
2. P. Bratley & J. McKay, "More amicable numbers," Math. Comp., v. 22, 1968, pp. 677678. MR 37 #1299.
3. E. B. Escorr, "Amicable numbers," Scripta Math., v. 12, 1946, pp. 61-72. MR 8, 135.
4. M. Garcia, "New amicable pairs," Scripta Math., v. 23, 1957, pp. 161-171. MR 20 #5158.
5. P. Hagis, Jr.,
appear.)
6. H.-J. Kanold,
"Relatively
prime amicable numbers
"Untere Schranken
1953, pp. 399-401. MR 15, 400.
für teilerfremde
of opposite parity,"
befreudete
7. E. J. Lee, "Amicable numbers and the bilinear diophantine
1968, pp. 181-187. MR 37 #142.
8. P. Poulet,
Math. Mag. (To
Zahlen," Arch Math., v. 4,
equation,"
Math. Comp., v. 22,
"43 new couples of amicable numbers," Scripta Math., v. 14, 1948, p. 77.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use