possibly, difference over interval using NumberPrimesLegendreModified (related previous ticket cd42b3ddf8,2017-05-28 )
would be useful to some. two numbers entered, limit1 and limit2 and difference returned. both limits must be greater than one. find interval of est. primes from two entries on positive number line. Inclusive, interval differences for numberPrimesGauss, numberPrimesLegendre, NumberPrimesLegendreModified might be useful.
##
### differenceNumberPrimesLegendreModified --
#     Return the approximate difference number of primes
#     between a lower and higher limit as given values
#     for approximate number of primes based on the
#     modified formula by Legendre
#
# Arguments:
#     limit1     The lower limit for the interval, largest prime to be included in the l.limit
#     limit2     The upper limit for the interval, largest prime to be included in the u.mlimit
#
# Returns:
#     Approximate difference number of primes
#
proc ::math::numtheory::differenceNumberPrimesLegendreModified {limit1 limit2} {
    if { $limit1 <= 1 } {
        return -code error "The lower limit must be larger than 1"
    }
    if { $limit2 <= 1 } {
        return -code error "The upper limit must be larger than 1"
    }

     set aa [::math::numtheory::numberPrimesLegendreModified [expr ($limit1)]]
     set bb [::math::numtheory::numberPrimesLegendreModified [expr ($limit2)]]
     expr ( $bb-$aa)
}
#######
statements invoking difference
 puts " [ ::math::numtheory::differenceNumberPrimesLegendreModified 100  200  ] "
         #puts " [ ::math::numtheory::differenceNumberPrimesLegendreModified 100  0  ] " invokes error.