Anomalous cancellation is a "canceling" of digits of 
and 
in the numerator and denominator
of a fraction 
which results in a fraction equal to the original. Note
that if there are multiple but differing counts of one or more digits in the numerator
and denominator there is ambiguity about which digits to cancel, so it is simplest
to exclude such cases from consideration.
There are exactly four anomalous cancelling proper fractions having two-digit base-10 numerator and denominator:
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(1)
|
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(2)
|
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(3)
|
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(4)
|
(c.f. Boas 1979). The first few 3-digit anomalous cancelling numbers are
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(5)
|
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(6)
|
and the first few with four digits are
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(7)
|
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(8)
|
The numbers of anomalously cancelling proper fractions having 
digits in both numerator and denominator for 
, 2, ... are 0, 4, 161, 1851, ....
The numbers of anomalously cancelling proper fractions having 
or fewer digits in both numerator and denominator for
,
2, ... are 0, 4, 190, 2844, ....
The concept of anomalous cancellation can be extended to arbitrary bases. For two-digit anomalous cancellation, anomalously cancelling fractions correspond to solutions to
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(9)
|
for integers 
.
Prime bases 
have no two-digit solutions, but there is a solution corresponding
to each proper divisor of a composite 
.
When 
is prime, this type of solution is the only one.
For base 4, for example, the only two-digit solution is 
. The number of solutions is even
unless 
is an even square.
Boas (1979) gives a table of solutions for 
. For the first few composite bases 
, 6, 8, 9, ..., the numbers of two-digit solutions are 1,
2, 2, 2, 4, 4, 2, 6, 7, 4, 4, 10, 6, 6, 6, 4, 6, 10, 6, 4, 8, 6, 6, 21, 2, 6, ...
(OEIS A259981).
