I'm going to start with a few examples. I may need someone to help correct wording.
I'm going to write what I call the modular fingerprint of the following numbers. It's the list of the remainders of these numbers when divided by all primes smaller than the number itself:
$15\equiv$
$1\pmod{2}$, $0\pmod{3}$, $0\pmod{5}$, $1\pmod{7}$, $4\pmod{11}$, $2\pmod{13}$
So 15's list would be $\{1,0,0,1,4,2\}$.
17 -- $\{1,2,2,3,6,4\}$
51 -- $\{1,0,1,2,7,12,0,13,5,22,20,14,10,8,4\}$
Question 1:
Are there infinitely many numbers that begin with any certain fingerprint? For instance, there are infinite that begin with $\{1\}$: all the odds. There are infinite that begin with $\{1,0\}$: odd multiples of $3$. Are there also infinite numbers that begin with the same fingerprint as $51$?
Question 2:
If #1 is true, are these fingerprints continuous? For instance, is there at least one (and thus infinitely many) number for each possible starting fingerprint?
$\{0\},$
$\{0,0\}, \{0,1\}, \{0,2\}, \{1,0\}, \{1,1\}, \{1,2\},$
$\{0,0,0\}, \{0,0,1\}, \{0,0,2\}, \{0,0,3\}, \{0,0,4\}, \{0,1,0\}, \{0,1,1\}, \ldots, \{1,2,4\}$, etc.
Key phrase: Chinese remainder theorem.