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COUSINS OF SMITH NUMBERS: MONICA AND SUZANNE SETS
Michael Smith
5400 N. Wayne, Chicago IL 60640
(Submitted February 1994)
1. INTRODUCTION
For any x eM, x may be expressed as the sum ao+a^lO 1 )*--- + ^(10^), where each
a,E{0,l,2,...,9}.
Suppose that x GN is a composite number. Then x = PiP2-Pm, where each pk is a prime
number. We can then formally define two functions S(x) and Sp(x) as
d
S(x) = ^aJ
m
and
S^^Sif,).
That is, S{x) is the digital sum of x and Sp(x) is the digital sum of the prime factors of x.
Wilansky [2] defines a Smith number as a composite integer where S(x) = Sp(x). This paper
deals with two kinds of sets related to Smith numbers. These sets are called Monica sets and
Suzanne sets.
Definition 1.1: The rfi1 Monica set M.w consists of those composite numbers x for which
n\S(x)-Sp(x) [we write S(x) =nSp(x)].
Definition 2.1: The rfr Suzanne set S„ consists of those composite numbers x for which n\S(x)
and n\Sp(x).
It should be noted that because I developed this concept from Smith numbers, I consider it to
be akin to Smith's. Therefore, I have named these sets after my cousins, Monica and Suzanne
Hammer.
2. ON THE POPULATION OF MONICA AND SUZANNE SETS
The following theorems give indications of what sort of integers belong to Monica and
Suzanne sets.
Theorem 2.1: If x is a Smith number, then x GM„, \/n eN.
Theorem 2.2: x e S„ => x eMn.
Note that the converse of Theorem 2.2 is not true; for example, 10 = 5x2, thus £(10) = 1 and
£,(10) = 7. 10 e M 6 since 6|l-7, but 10 £S 6 since 6|1.
Theorem 2.3: For any integer k > 1, if x is a k-Smith, then x G M H .
Proof: McDaniel [1] defines a k-Smith as a composite number x such that kS'(x) = Sp(x).
Thus, S(x) - Sp(x) is divisible by k - 1 . Therefore, x eMk_x. D
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COUSINS OF SMITH NUMBERS: MONICA AND SUZANNE SETS
3. EPILATIONS BETWEEN SETS OF MONICAS AND SETS OF SUZANNES
There are some rather simple properties of Monica and Suzanne sets that may be useful in
later studies.
Theorem 3:
(a) If p, q
(b) If py q
(c) If p, q
(d) If p, q
GN and p\q,
GH and p\q,
GN and p\q
GN and p\q
then x GMq implies x GMp;
then x eSq implies x eS^;
are relatively prime, then x GMp and x sMq implies x G M W ;
are relatively prime, then x sSp and x e$q implies x ^Spq.
4. INFINITE ELEMENTS IN EACH MONICA AND SUZANNE SET
The most interesting property of Monica and Suzanne sets is that every Monica set and every
Suzanne set has an infinite number of elements. McDaniel [1] proves that there is an infinite
number of Smiths; this implies, by Theorem 2.1, that every Monica set has an infinite number of
elements.. The proof that there is an infinite number of elements in each Suzanne set is more
complicated.
Theorem 4: All Suzanne sets have infinitely many elements.
Proof: Consider Sv For any composite number x, l\S(x) and l\Sp(x).
For S„, where n > 1, we need to construct a candidate integer r such that S(r) = n. Let r be
an w-digit Repunit, that is, a string of n ones (see [3]). Let z = ar, where a is determined by the
following table:
then a = l since SpQr) = Sp(r)
Sp{r)=70
then a = 9 since Sp(9) = 6
Sp(r)^l
Sp(r)^72
then a~5 since Sp(S) - 5
5 p (r) S ,3 then a = 4 since S (4) = 4
then a = 3 since S (3) = 3
Sp(r)^4
Sp(r)^75 then a = 2 since Sp(2) = 2
then a = 15 since Sp(\5) = 8 = 7 1
Sp{r)^6
From the table it should be obvious that 7\Sp(r) + Sp(a), and thus 7\Sp(z).
S(z) = S(r)S(a) because of our choice of r, so n\S(z).
Note that
Let m be an integer such that n\(Sp(z) + 7m) and let y - z * 10w. Clearly, Sp(y) - Sp(z) +
Sp(\0m) and Sp(\0m) = 7m; thus, n\Sp(y) = Sp(z) + Sp(lOm).
Note that S(y) = S(z), so n\S(y); thus, a r * 10w =y GS„ for all m such that n\Sp(ar) + 7m.
Clearly, there are infinitely many choices for m. •
ACKNOWLEDGMENTS
I would like to thank my brother Matthew for his (nontechnical) assistance, Dr. Samuel
Wagstaff at the Department of Computer Science at Purdue University for his most helpful
1996]
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COUSINS OF SMITH NUMBERS: MONICA AND SUZANNE SETS
suggestions, and Sepultura for the music video I watched when I came up with the linchpin of the
proof for Theorem 4.
REFERENCES
1. W. McDaniel. "The Existence of Infinitely Many £-Smith Numbers." The Fibonacci Quarterly 25.1 (1987):76-80.
2. A. Wilansky. "Smith Numbers." Two Year College Math Journal 13 (1982):21.
3. Samuel Yates. "The Mystique of Repunits." Math Mag. 51 (1978):22-28.
AMS Classification Numbers: 11A63, 11A07
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