So, I remember a long time ago i found a thread that had this formula to finding your birthday that corresponds to that pokemons number. sadly, there are 493 pokemon and only 365 days. but there was one where it included all of the pokemon; which i can not recall.....

so here is what you do:

-count from January 1st all the days until you reach your birthday.
-take that number down.
-go to the national dex and find the pokemon that shares the same number as the total of days from January 1st.

example:

-my birthday is March 25th
-count all the days from January 1st to March 25th
-31 days in January + 28 days in February + 25 days into March = 84
-the pokemon with the number 84 is Doduo; my birthday pokemon is Doduo

example2:

-my sisters birthday is July 2nd
-count all the days from January 1st to July 2nd
-31 days in January + 28 days in February + 31 days in March + 31 days in April + 30 days in May + 30 days in June + 2 days into July = 183
-the pokemon with the number 183 is marill; my sisters birthday pokemon is
marill

hope this works;3

These are just for fun, they have nothing to do with the actual birthday pokemon. its just using the number you got for fun.

i came up with just a few other ones that are possible to do as well. this one is where you take the number that you already calculated, mine being 84, and look at 184, 284, 384, and 484 too.
Giving me:
it kind of includes the Sinnoh pokemon when you do this one. just put a 4 in front of the last 2 digits of your number. if your number's digits are from 494-499 when you add the 4 then just subtract 10 until you get to an existing pokemon's number.

Doduo
Azumarill
Masquerain
Rayquaza minus 10-->Beldum
Palkia minus 10-->Porygon-Z
(if you want to stay clear of legendaries minus 10 until you get to regular pokemon)

also trying switching the last two digits in your number to see what you get(these are all just random stuff you can doXD)

so 48, 148, 248, 348, 448. lets see what i got:

Venonat
Dragonair
Tyranitar
Armaldo
Lucario

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ALSO! thanks to Android Raptor and Sleet, they have come up with two different ways for getting your sinnoh and and birthday pokemon(this one includes sinnoh as a whole.)

Android Raptor:

Just multiply the month of your birth times the day, then just put together (dont add) the last two digits of the year. I'll use mine (March 25, 1993) as an example:

3(March) X 25(Day) = 75 + 93(last two digits of year)= 168

which would make me a...........Ariados! im not complaining

Sleet:

this way is sort of the same. March 25, 1993:
-add the number of days from january 1st to your bday, mine is 84, to the day of the month, mine is 25, and add that to the number of the week day it falls on (sunday=1, monday=2, etc.), 25 is on a wednesday which is the 4th day of the week.
-then times it by the total year, mine is 1993
-put it all together and most likely you will get a largeer number than 493. now just keep subtracting 493 until you get to a pokemons number.
-Good Luck!

84 + 25 + 4 = 113 x 1993 = 22509 - 493 until you get to a number less than or equal to 493 = 324 which is Torkoal!

a faster way, if you have a really big number, is to minus 5423 which is (493 x 11)until you get to something close to or less than 5423, and then continue to subtract 493.

NOTE: this will work out differently every year because the number always changes to a different week day every year