I am definitely into things I don't understand here. :(

#!/usr/bin/perl -w
use strict;

use bignum;
use Digest::SHA qw(sha256_hex);

# a small expansion on the previous code block
for (my $i = 1; $i <= 9; $i++)
    {
    my $digest = hex(sha256_hex($i));

print "$i $digest\n";
$digest /= 64;
print "$i $digest\n";
    }


1 48635463943209834798109814161294753926839975257569795305637098542720
+658922315
1 75992912411265366872046584627023053010690000000000000000000000000000
+0000000
2 96094161643976066833367867971426158458230048495430276217795328666133
+331159861
2 15014712756871260442713729370535337259100000000000000000000000000000
+00000000
3 35293215426786447154857697798367884701614677727176325092965345248689
+205321678
3 55145649104353823679465152809949819846270000000000000000000000000000
+0000000
4 33984360982413536682390860969296307922929415152052354251133793603654
+468157322
4 53100564035021151066235720264525481129580000000000000000000000000000
+0000000
5 10818240655469999731486825095773005325907358940236581977057259273658
+4172823453
5 16903501024171874580448164212145320821730000000000000000000000000000
+00000000
6 10492023800686533720501340709024820017001830686534338836405100876796
+5015414403
6 16393787188572708938283344857851281276570000000000000000000000000000
+00000000
7 54734117258892461880478870895348100103198302433611928089023761078319
+507514449
7 85522058217019471688248235773981406411250000000000000000000000000000
+0000000
8 20075373234943686845167158285967784892467090849631486320124245130906
+619831459
8 31367770679599510695573684821824663894480000000000000000000000000000
+0000000
9 11463511883541201508393500803711559652562721115888819482889049505863
+143503287
9 17911737318033127356864845005799311957130000000000000000000000000000
+0000000
[download]

I start getting incorrect mathematical calculations here.

#!/usr/bin/perl -w
use strict;

# changed only this line
use bignum (p => 1);
use Digest::SHA qw(sha256_hex);

for (my $i = 1; $i <= 9; $i++)
    {
    my $digest = hex(sha256_hex($i));

print "$i $digest\n";
$digest /= 64;
print "$i $digest\n";
    }


1 48635463943209834798109814161294753926839975257569795305637098542720
+658922315
1 81059106572016391330183023602157923211399958762616325509395164237867
+7648705
2 96094161643976066833367867971426158458230048495430276217795328666133
+331159861
2 16015693607329344472227977995237693076371674749238379369632554777688
+88852664
3 35293215426786447154857697798367884701614677727176325092965345248689
+205321678
3 58822025711310745258096162997279807836024462878627208488275575414482
+0088695
4 33984360982413536682390860969296307922929415152052354251133793603654
+468157322
4 56640601637355894470651434948827179871549025253420590418556322672757
+4469289
5 10818240655469999731486825095773005325907358940236581977057259273658
+4172823453
5 18030401092449999552478041826288342209845598233727636628428765456097
+36213724
6 10492023800686533720501340709024820017001830686534338836405100876796
+5015414403
6 17486706334477556200835567848374700028336384477557231394008501461327
+50256907
7 54734117258892461880478870895348100103198302433611928089023761078319
+507514449
7 91223528764820769800798118158913500171997170722686546815039601797199
+1791907
8 20075373234943686845167158285967784892467090849631486320124245130906
+619831459
8 33458955391572811408611930476612974820778484749385810533540408551511
+0330524
9 11463511883541201508393500803711559652562721115888819482889049505863
+143503287
9 19105853139235335847322501339519266087604535193148032471481749176438
+5725055
[download]

And here, things have really gone off of the rails.

#!/usr/bin/perl -w
use strict;

use bignum (p => 1);
use Digest::SHA qw(sha256_hex);

# here, I change 9 to 10
for (my $i = 1; $i <= 10; $i++)
    {
    my $digest = hex(sha256_hex($i));

print "$i $digest\n";
$digest /= 64;
print "$i $digest\n";
    }


1 48635463943209834798109814161294753926839975257569795305637098542720
+658922315
1 81059106572016391330183023602157923211399958762616325509395164237867
+7648705
2 96094161643976066833367867971426158458230048495430276217795328666133
+331159861
2 16015693607329344472227977995237693076371674749238379369632554777688
+88852664
3 35293215426786447154857697798367884701614677727176325092965345248689
+205321678
3 58822025711310745258096162997279807836024462878627208488275575414482
+0088695
4 33984360982413536682390860969296307922929415152052354251133793603654
+468157322
4 56640601637355894470651434948827179871549025253420590418556322672757
+4469289
5 10818240655469999731486825095773005325907358940236581977057259273658
+4172823453
5 18030401092449999552478041826288342209845598233727636628428765456097
+36213724
6 10492023800686533720501340709024820017001830686534338836405100876796
+5015414403
6 17486706334477556200835567848374700028336384477557231394008501461327
+50256907
7 54734117258892461880478870895348100103198302433611928089023761078319
+507514449
7 91223528764820769800798118158913500171997170722686546815039601797199
+1791907
8 20075373234943686845167158285967784892467090849631486320124245130906
+619831459
8 33458955391572811408611930476612974820778484749385810533540408551511
+0330524
9 11463511883541201508393500803711559652562721115888819482889049505863
+143503287
9 19105853139235335847322501339519266087604535193148032471481749176438
+5725055
10 3359281535163583886622750554468341756416232122807515421831543601247
+6449103317
10 5598802558605973144371250924113902927360386871345859036385906002079
+40818389
10 3359281535163583886622750554468341756416232122807515421831543601247
+6449103317
10 5598802558605973144371250924113902927360386871345859036385906002079
+40818389
10 3359281535163583886622750554468341756416232122807515421831543601247
+6449103317
10 5598802558605973144371250924113902927360386871345859036385906002079
+40818389

[repeats indefinitely]
[download]

I am definitely into things I don't understand here

I, too, don't know why it repeats indefinitely, but I can confirm that I see the same behaviour.

If you want something that DWIMs nicely, and works quickly and reliably, I would suggest using either Math::GMPz or Math::GMP.
Either will do, though they both require the GMP C library:

#!/usr/bin/perl -w
use strict;

use Math::GMPz;
# use Math::GMP;
use Digest::SHA qw(sha256_hex);

for (my $i = 1; $i <= 10; $i++) {
  my $digest = Math::GMPz->new(sha256_hex($i), 16);
# my $digest = Math::GMP-> new(sha256_hex($i), 16);
  print "$i $digest\n";
  $digest /= 64;
  print "$i $digest\n";
}
[download]

Output is:

1 48635463943209834798109814161294753926839975257569795305637098542720
+658922315
1 75992912411265366872046584627023053010687461339952805165057966473001
+0295661
2 96094161643976066833367867971426158458230048495430276217795328666133
+331159861
2 15014712756871260442713729370535337259098445077410980659030520104083
+33299372
3 35293215426786447154857697798367884701614677727176325092965345248689
+205321678
3 55145649104353823679465152809949819846272933948713007957758351951076
+8833151
4 33984360982413536682390860969296307922929415152052354251133793603654
+468157322
4 53100564035021151066235720264525481129577211175081803517396552505710
+1064958
5 10818240655469999731486825095773005325907358940236581977057259273658
+4172823453
5 16903501024171874580448164212145320821730248344119659339151967615091
+27700366
6 10492023800686533720501340709024820017001830686534338836405100876796
+5015414403
6 16393787188572708938283344857851281276565360447709904431882970119994
+53365850
7 54734117258892461880478870895348100103198302433611928089023761078319
+507514449
7 85522058217019471688248235773981406411247347552518637639099626684874
+2304913
8 20075373234943686845167158285967784892467090849631486320124245130906
+619831459
8 31367770679599510695573684821824663894479829452549197375194133017041
+5934866
9 11463511883541201508393500803711559652562721115888819482889049505863
+143503287
9 17911737318033127356864845005799311957129251743576280442014139852911
+1617238
10 3359281535163583886622750554468341756416232122807515421831543601247
+6449103317
10 5248877398693099822848047741356783994400362691886742846611786876949
+44517239
[download]

Cheers,
Rob