[ June 4, 2023 ] [ Last update September 30, 2023 ]
Building a relationship between the natural numbers
and the ninedigitals using the function
"Prepending n digits d to k so that [d^^n][k]
becomes a record delayed prime"
Patrick De Geest

When I use the term ninedigital in this plate I always refer to
a strictly zeroless pandigital (digits from 1 to 9 each appearing just once).

( '^^' is symbol for concatenation )

This plate is in fact an offshoot from the material presented in
WONplate 197 (appending digits to k) & more in particular
WONplate 224 (prepending digits to k). So my advise to get
familiarized with the topic presented in those plates.

In plate 197 one can only append digits 1, 3, 7 and 9 to arrive at (probable) primes
whereas in WONplate 224 one can prepend all the nine digits to k. So we will
continue only with the latter because that gives already a clue for arriving
at our ninedigitals. All will become clear with a couple of examples.

Example 1
Start eg. with prepending 50 digits d's (n=50) to k and find its value so that their
concatenation yield a prime or PRP with the extra condition that all smaller numbers n
from 1 to 49 (n-1) yield only composites. Formally we present this as [d^^n][k].
Now let us replace d with the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9. We get:

Example 2
This time our seed will be 'the number of the beast' 666.

Example 3
Let us search for the ninedigital of the year 2023.

Here is some homework for my readers. Beware that it might be hard
as high seeds needs lots of time to calculate. But PRP's are accepted.

1. Find a ninedigital duo with 2 or more seeds.

\\ This is my PARI/GP program code for WONplate 225 by PDG [June 9, 2023] → seq225.gp
{
seed=40;
m=matrix(9,2);
sumk=0;
for(d=1, 9,
     for(a=seed, seed,
          b=1;
          until(flag==0,
               f=(d*(10^a-1)/9)*10^(#(digits(b)))+b;
               while(ispseudoprime(f)==0,
                    b+=2; f=(d*(10^a-1)/9)*10^(#(digits(b)))+b;
               );
               ad=0; flag=0;
               for(i=1, a-1,
                    ad++; fd=(d*(10^ad-1)/9)*10^(#(digits(b)))+b;
                    if(ispseudoprime(fd), flag++; break(); );
               );
               b+=2;
          );
          k=b-2; m[d,1]=k; m[d,2]=d; sumk+=k;
          print("[",d,"^^",a,"][",k,"]");
          b=1; 
     );
);
v=vecsort([m[1,1],1; m[2,1],2; m[3,1],3; m[4,1],4; m[5,1],5; m[6,1],6; m[7,1],7; m[8,1],8; m[9,1],9]~);
print(v);
print("Seed ",seed," yield ninedigital ",v[2,1],v[2,2],v[2,3],v[2,4],v[2,5],v[2,6],v[2,7],v[2,8],v[2,9]);
print("Sum of k's = ",sumk);
cnt=0; print1("{ "); for(i=1,9, if(isprime(v[1,i]),cnt++; print1(v[1,i]," "))); print1("} ");
print("Primescore is ",cnt,"/9");
}