Can we expect infinite many primes $p$ equal to the start of $frac(\sqrt{p})$?

Question

The following routine searches for prime numbers $\ p\ $ equal to the beginning of the fractional part of the decimal expansion of $\ \sqrt{p}\ $ :

? forprime(p=1,4*10^9,s=length(digits(p));x=frac(sqrt(p));if(truncate(x*10^s)==p
,print1(p," ")))
5711 8053139
?

• Can we expect infinite many primes with the given property ?
• What is the next prime number with the given property ?