The celebrated Lucas-Lehmer test it is a deterministic and very fast test to determine the primality of Mersenne numbers, numbers in the family \(\{M_n=2^n-1\}\) (or more generally numbers in the family \(\{A2^n-1\},\, A\) an odd number, \(A<2^n\)). The test is based on the properties of the Lucas sequences \(\{U_n\},\, \{V_n\}\) corresponding to the roots of a quadratic polynomial \(x^2-Px-Q,\, P,Q\in \mathbf{Z}\). The Lucas-Lehmer test has been generalized to many others families of numbers, see [H. C. Williams, Édouard Lucas and primality testing. New York, NY: John Wiley & Sons (1998; Zbl 1155.11363)].
Following these steps the present paper provides two Lucas-Lehmer-like algorithms to test the primality of numbers in the family \(\{N_d(n,m)=10^{2n+m} -d10^n -1\},\, m\) odd, \(d\) a number of \(m\) digits. The authors argue that those numbers are of interest in recreational mathematics, since they can be prime curios, see [J. N. Friend, Numbers: fun and facts. New York: Charles Scribner’s Sons (1954; Zbl 0059.00102)], primes with a peculiar shape (in base 10 representation), for instance palindrome numbers, numbers with shape \(99\cdots 9,g,99\cdots 9,g,99\cdots, 9\), etc.
Sections 2 and 3 recall the Lucas-Lehmer test and some more general class of tests based on linear recurrence sequences.

The first algorithm for numbers \(N_d(n,m)\) is given as Corollary 1 (of Theorem 5) in Section 4 while the second is given in Corollary 3 (of Theorem 12) and Corollary 4 (of Theorem 13) in Section 7. Both tests assume it is known a prime \(q,\, q\equiv 1 \bmod 5,\, N^{(q-1)/5} \neq 0,1 \bmod q\).

Finally Section 8 shows some computational results (Tables 1 to 12). Tables 1 and 2 give the smallest \(q\) verifying the above condition for \(N \in N_d(n,1)\) and \(3\le n \le 11000\). The rest of Tables give prime numbers and run time for some \(N\) in the families \(N_d(n,i),\, i=1,3,5,7,9\).