Two conjectures of B. R. Santos concerning totitives

Abstract:

Recently B. R. Santos conjectured that 12 is the largest integer n with the following property: $(\ast )$ \[ \left \{ {\begin {array}{*{20}{c}} {\begin {array}{*{20}{c}} {{\text {If}}\;m\; \in \;[1,n]\;{\text {and}}\;n\;{\text {are}}\;{\text {relatively}}\;{\text {prime,}}\;{\text {then}}} \hfill \\ {n + m\;{\text {is}}\;{\text {a}}\;{\text {prime}}\;{\text {number}}{\text {.}}} \hfill \\ \end {array} } \hfill \\ \end {array} } \right .\] Using deep numerical estimates of Rosser and Schoenfeld for the number $\pi (x)$ of primes less than x, it is proved that the conjecture of Santos is true. The same result holds, if in addition it is assumed in $(\ast )$ that m is a prime.