Problems & Puzzles: Puzzles

 Puzzle 941. nxtprime(p) inside p^2 Carlos Rivera poses the following puzzle: Regarding this statement: "Primes whose square has the next prime as substring', is already known (see A052073): 23, 83, 113, 1123, 200003, 328127, 381289, 714597769, 4916552822383", in a subsequent comment to this sequence we may read: "The prime 482564152712479922509389813571 is also a term. -Giovanni Resta, May 24 2018" Q1. Can you guess the method that Giovanni used to found this large prime? Q2. If you get the method asked in Q1, can you produce a second prime like that?

Contribution came from Emmanuel Vantieghem.

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Emmanuel wrote:

Q1.
I guess Giovanni could have worked along the following lines :
Let's search for numbers  m  (not necessarily prime) for which there exist a number  g  such that the number  m+g  is 'visible' in  m^2.
Say, on the left of some  k-digit number  b.
If  n  is the number of digits of  m  then we should have :
m^2 = a*10(n+k) + (m+g)*10^k + b,
If we fix some values for  k  and  g, we can try to find  m  from the equation :

m^2 - m*10^k -  g*10^k - b === 0 (mod 10^(n+k))  for some  k-digit  b.
This is a modular quadratic equation of type
Ax^2 + 2Bx + C === 0 (mod M)
whose solutions are  x === -B + s, x === -B - s (mod M),
provided  s  is a number such that  s^2 === B^2 - AC (mod M).
This is of course possible if and only if  B^2 - AC  is a quadratic residue modulo M.
So, we must look for a value  b  which makes the number
d = 5*10^(2k-2) + g*10^k + b
congruent to a square  s^2 mod 10^(n+k).  We then only must see if  5*g*10^(2k-2)+s (or - s)  has indeed  n  digits.

I illustrate this by the example that finally lead to Giovanni's prime number.
We take  k = 4  and  g = 6.  So, we try to solve the congruence
m^2 - 10000m - 60000g + b == 0 (mod  10n+4).
for some  b  and some  n.
When we take  b  in the set of quadratic residues modulo  10000 (which is easy to compute with the aid of some mathematical program),
we will be 'lucky' for some values of  n.
For instance, if you take  b = 1, you find  m = 437474999 (not prime); m+g = 437475005 (also not prime)
and indeed : m^2 = 191384374750050001.
Whith a littlebit of patience, we find Giovanni's prime for  b = 2041  and  n = 30.

Q2.
I tried many values for  k, g, b, n.
Unfortunately,primality for  m  was very rare, primality for m+g  was even more rare : most rare was when  m+g  was the nextprime of  m.
Finally I found the prime  p = 9433515143067535217 when  k = 3, g = 20, n = 19.
NextPrime(p) =  9433515143067535237 ; p^2 = 88991207954484499433515143067535237089.
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If this can be useful : here is the Mathematicapromgram that I used (I only had to choose the value k) :

k=3;B=Select[Range[1,10^k-1,2],Mod[#,8]==1&&JacobiSymbol[#,5^k]==1&];B=Select[B,Mod[#,5]!=0&];Lb=Length[B]
g=0;While[g<20,g=g+2;
i=1;While[i<Lb,i++;b=B[[i]];n=12;While[n<31,n++;d=25 10^(2k-2)+10^k g+b;M=Select[5 10^(k-1)+PowerModList[d,1/2,10^(n+k)],IntegerLength[#]==n&];If[M!={},
m=M[[1]];If[PrimeQ[m],If[PrimeQ[m+g],Print[g,"  ",b,"  ",n,"  ",m,"  ",m+g," : ",NextPrime[m]==m+g]]]]]]]

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