A Negative Answer to Two Questions about the Smallest Prime Numbers Having Given Digital Sums

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. Moreove... View More