In this paper I make the following four conjectures: (I) there exist always a prime of the form p^2 + 18*m between the squares p^2 and q^2 of a pair of twin primes [p, q = p + 2], beside the pair [3, 5]; examples: for [p^2, q^2] = [11^2, 13^2] = [121, 169] there exist the primes 139 = 121 + 1*18) and 157 = 121 + 2*18; for [p^2, q^2] = [17^2, 19^2] = [289, 361] there exist the prime 307 = 289 + 1*18); (II) there exist always a prime of the form q^2 – 18*n between the squares p^2 and q^2 of a pair of twin primes [p, q = p + 2], beside the pair [3, 5]; examples: for [p^2, q^2] = [11^2, 13^2] = [121, 169] there exist the prime 151 = 169 – 1*18); for [p^2, q^2] = [17^2, 19^2] = [289, 361] there exist the prime 307 = 361 – 3*18); (III) there exist an infinity of r primes of the form p^2 + 18*m or q^2 – 18*n between the squares p^2 and q^2 of a pair of twin primes [p, q = p + 2] such that the number obtained concatenating p^2 to the right with r is prime; example: 121139 is prime; (IV) there exist an infinity of r primes of the form p^2 + 18*m or q^2 - 18*n between the squares p^2 and q^2 of a pair of twin primes [p, q = p + 2] such that the number obtained concatenating q^2 to the left with r is prime; example: 139169 is prime. Of course, the conjectures (III) and (IV) imply that there exist an infinity of pairs of twin primes.