A NEGATIVE ANSWER TO TWO QUESTIONS ABOUT THE SMALLEST PRIME NUMBERS HAVING GIVEN DIGITAL SUMS
2015
Denote by ρ(k) the smallest prime number with digital sum k (not a multiple of 3). Richard K. Guy asked whether the congruences ρ(k) ≡ 99 (mod 100) and ρ(k) ≡ 999 (mod 1000) hold for all k> 38, respectively k> 59. Counterexamples to this are given, inter alia, for k = 86 and k = 104. Moreover, several open problems and conjectures about ρ(k) are discussed. 1.