<s>
In	O
number	O
theory	O
,	O
a	O
perfect	B-Language
digit-to-digit	I-Language
invariant	I-Language
(	O
PDDI	B-Language
;	O
also	O
known	O
as	O
a	O
Munchausen	B-Language
number	I-Language
)	O
is	O
a	O
natural	O
number	O
in	O
a	O
given	O
number	O
base	O
that	O
is	O
equal	O
to	O
the	O
sum	O
of	O
its	O
digits	O
each	O
raised	O
to	O
the	O
power	O
of	O
itself	O
.	O
</s>
<s>
An	O
example	O
in	O
base	O
10	O
is	O
3435	B-Language
,	O
because	O
.	O
</s>
<s>
The	O
term	O
"	O
Munchausen	B-Language
number	I-Language
"	O
was	O
coined	O
by	O
Dutch	O
mathematician	O
and	O
software	O
engineer	O
Daan	O
van	O
Berkel	O
in	O
2009	O
,	O
as	O
this	O
evokes	O
the	O
story	O
of	O
Baron	O
Munchausen	O
raising	O
himself	O
up	O
by	O
his	O
own	O
ponytail	O
because	O
each	O
digit	O
is	O
raised	O
to	O
the	O
power	O
of	O
itself	O
.	O
</s>
<s>
A	O
natural	O
number	O
is	O
defined	O
to	O
be	O
a	O
perfect	B-Language
digit-to-digit	I-Language
invariant	I-Language
in	O
base	O
b	O
if	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
number	O
3435	B-Language
is	O
a	O
perfect	B-Language
digit-to-digit	I-Language
invariant	I-Language
in	O
base	O
10	O
because	O
.	O
</s>
<s>
for	O
all	O
,	O
and	O
thus	O
1	O
is	O
a	O
trivial	O
perfect	B-Language
digit-to-digit	I-Language
invariant	I-Language
in	O
all	O
bases	O
,	O
and	O
all	O
other	O
perfect	B-Language
digit-to-digit	I-Language
invariants	I-Language
are	O
nontrivial	O
.	O
</s>
<s>
For	O
the	O
second	O
convention	O
where	O
,	O
both	O
and	O
are	O
trivial	O
perfect	B-Language
digit-to-digit	I-Language
invariants	I-Language
.	O
</s>
<s>
A	O
perfect	B-Language
digit-to-digit	I-Language
invariant	I-Language
is	O
a	O
sociable	O
digit-to-digit	O
invariant	O
with	O
.	O
</s>
<s>
This	O
means	O
also	O
that	O
there	O
are	O
a	O
finite	O
number	O
of	O
perfect	B-Language
digit-to-digit	I-Language
invariant	I-Language
and	O
cycles	O
for	O
any	O
given	O
base	O
.	O
</s>
<s>
The	O
following	O
program	O
in	O
Python	B-Language
determines	O
whether	O
an	O
integer	O
number	O
is	O
a	O
Munchausen	B-Language
Number	I-Language
/	O
Perfect	O
Digit	O
to	O
Digit	O
Invariant	O
or	O
not	O
,	O
following	O
the	O
convention	O
.	O
</s>
<s>
The	O
examples	O
below	O
implements	O
the	O
perfect	B-Language
digit-to-digit	I-Language
invariant	I-Language
function	O
described	O
in	O
the	O
definition	O
above	O
to	B-Algorithm
search	I-Algorithm
for	I-Algorithm
perfect	I-Algorithm
digit-to-digit	I-Algorithm
invariants	I-Algorithm
and	I-Algorithm
cycles	I-Algorithm
in	O
Python	B-Language
for	O
the	O
two	O
conventions	O
.	O
</s>
