Regarding Q.1 For D=2,3,5,6,7,8,
The Smallest Titanic Prime 10^999+7 is such that its square is free of
digit D.

For D=4,9 , The Titanic Prime
10^999+3561 is such that its square is free of digit D.

A Titanic prime such that its
square is free of D for D=1. This Titanic prime is: 2*10^999+8567

... the solution for D=0 can not be
attempted without evolving some strategy to tackle most probable
solutions only. Lets try and hope to find a solution.

I have used Primeform/GW to find
solutions and Primo to prove a few primes. Numbers with special decimal
forms are requested and I have found the solutions by experimenting with
such numbers and not much analysis.

**1. **Find one titanic prime
(digits=>1000) such that its square is free of the digit D, for each of
the following values for D: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

The digits here are covered by the
following cases but my first test was: p = 10^999+7 is well-known as the
smallest titanic prime. p^2 is free of the digits 2, 3, 5, 6, 7, 8.

p = 9*10^1058+1 is easily proven
prime by Primeform with p-1 factored. p^2 is free of the digits 2, 3, 4,
5, 6, 7, 9.

p = 3*10^1137+7 is proven prime with
Primo.

p^2 is free of the digits 1, 3, 5, 6,
7, 8.

p = (10^1000-1)/3+49*10^132 is proven
prime with Primo.

p has exactly 1000 digits and p^2 is
only free of the digit 0. p also answers question 2.

The 3 previous primes are my own
finds covering all digits.

I also wanted a really big prime and
looked at the top 5000 list at primes.utm.edu. I did not have to look far.
Number 98 (list dated 18 Sep 2002): p = 105994*10^105994+1 is a
106000-digit Generalized Cullen prime found by **Guenter Loeh **and**
Yves Gallot** (presumably Loeh was running proth.exe). A quick test
shows that p^2 is missing the digit 5. I have not tested the bigger
numbers but would be extremely surprised to see a missing digit in a
square.

p = 10^69882+3*10^34941+1 with 69883
digits is the record palprime. It was found by **Daniel Heuer** with
PrimeForm. p^2 is free of the seven digits 2, 3, 4, 5, 7, 8, 9.

**2.** Find one titanic prime of
exactly 1000 digits such that its square is only free of the digit "zero"

(10^1000-1)/3+49*10^132 from question
1.

**3.** Find one titanic palprime
such that its square is free of the digit "zero"

p = (10^1867-1)/3+4*10^933 is a
probable prime.

p has 1867 digits, the middle is 7
and all others are 3.

p is a palprime - and also a
near-repdigit prime.

p^2 is free of the digit 0 (but also
of 4 and 6).

It would take around a day to prove p
prime with Primo.

I omitted the proof and proved the
next solution.

p = (10^1287-1)/3+313*10^642 is
proven prime with Primo.

p is palindromic and p^2 is only free
of the digit 0.

Let p(n) = (10^(2*n+3)-1)/3+313*10^n
for n>2

The previous solution is p(642).

p(n) is always palindromic and p(n)^2
is always free of only the digit 0.