What is the smallest prime $P>2$ with this property? Let P be a prime number and P=$m^{\text{th}}$ prime number. Here I'm interested in a prime P$>2$, such that P divides the concatenation of the first $(2m-1)$ prime numbers. Using my old laptop I've checked P up to $7919$ (i.e: up to the $1000^{\text{th}}$ prime number)without finding a solution. Can you find the smallest /some primes P$>2$ with such property ?

Update: Fixed typo (Sahra B.) and used Divisible (J.M.):
Is this the operation you mean?
concatenate[a__] := ToExpression[StringJoin[ToString /@ a]]

f[m_] := concatenate[Prime[#] & /@ Range[1, 2 m - 1]]

f[#] & /@ Range[5]


{2, 235, 235711, 2357111317, 23571113171923}

checkM[m_] := Divisible[f[m], Prime[m]]

Timing[Select[Range[2,2*10^6], checkM[#] &, 1]]

yields:

{0.207871, {303}}

suggesting that the 303th prime satisfies the above check?

You can look at code and actual integers involved [ here ].
