World!Of
Numbers

WON plate
224 |

[ May 20, 2023 ] [ Last update June 2, 2023 ]
Prepending n digits d to k so that [d^^n][k] becomes
a record delayed prime
Xinyao Chen (email)

While WONplate 197 is appending digits to the RIGHT of a number,
now let us consider prepending digits to the LEFT of the number.
Except for 0, all digits 1 to 9 can be used, but of course we only
consider the numbers k both coprime to 10 and coprime to d (since all
numbers [d^^n][k] are divisible by gcd(k,10) and gcd(k,d)).

Here are the record k-values for each case in base 10.

( '^^' is symbol for concatenation )

Also when numbers get larger I will accept PRP (PRobable Prime)
as valid entries.

Click on the header titles to open the related worksheets from Xinyao Chen.
The main page is at → https://sites.google.com/view/world-of-numbers-plate-219/

 BASE 10 Prime byprepending 1's Prime byprepending 2's Prime byprepending 3's [1^^1][1] [2^^3][1] [3^^1][1] [1^^17][11] [2^^204][17] [3^^2][19] [1^^713][33] [2^^19151][99] [3^^4][41] [1^^3372][1033] [2^^30843][737] [3^^7][127] [1^^35793][2233] [2^^34697][3717] [3^^8][157] [1^^ >100000][2661] [2^^ >100000][3737] [3^^34][443] [3^^24037][781] [3^^ >100000][4037] [1] to [1033] by Xinyao Chen[2233] & [2661] by PDG [1] to [99] by Xinyao Chen[737] to [3737] by PDG [1] to [443] by Xinyao Chen[781] to [4037] by PDG See alsoA200065 ```Sequence [k] in order of increasing prepended digits 1's. Each value < n in [d^^n][k] must be composite. ``` ```Sequence [k] in order of increasing prepended digits 2's. Each value < n in [d^^n][k] must be composite. ``` ```Sequence [k] in order of increasing prepended digits 3's. Each value < n in [d^^n][k] must be composite. ``` Prime byprepending 4's Prime byprepending 5's Prime byprepending 6's [4^^1][1] [5^^11][1] [6^^1][1] [4^^2][9] [5^^69][13] [6^^5037][11] [4^^15][11] [5^^182][61] [6^^ >100000][1859] [4^^50][27] [5^^240][181] [4^^187][141] [5^^645][241] [4^^837][319] [5^^1683][319] [4^^2846][609] [5^^79773][803] [4^^3112][1261] [5^^ >100000][1359] [4^^9688][1431] [4^^ >100000][1551] [1] to [1261] by Xinyao Chen[1431] & 1551 by PDG [1] to [319] by Xinyao Chen[803] & [1359] by PDG [1] to [11] by Xinyao Chen[1859] by PDG ```Sequence [k] in order of increasing prepended digits 4's. Each value < n in [d^^n][k] must be composite. ``` ```Sequence [k] in order of increasing prepended digits 5's. Each value < n in [d^^n][k] must be composite. ``` ```Sequence [k] in order of increasing prepended digits 6's. Each value < n in [d^^n][k] must be composite. ``` Prime byprepending 7's Prime byprepending 8's Prime byprepending 9's [7^^1][1] [8^^2][1] [9^^2][1] [7^^3][11] [8^^3][13] [9^^1798][13] [7^^18][29] [8^^77][33] [9^^3527][1177] [7^^64][79] [8^^96][149] [9^^8654][2587] [7^^173][123] [8^^11592][151] [9^^ >100000][2873] [7^^2502][293] [8^^68805][473] [7^^21186][821] [8^^89100][503] [7^^ >100000][909] [8^^ >100000][679] [1] to [293] by Xinyao Chen[821] & [909] by PDG [1] to [149] by Xinyao Chen[151] to [679] by PDG [1] to [1177] by Xinyao Chen[2587] & [2873] by PDG ```Sequence [k] in order of increasing prepended digits 7's. Each value < n in [d^^n][k] must be composite. ``` ```Sequence [k] in order of increasing prepended digits 8's. Each value < n in [d^^n][k] must be composite. ``` ```Sequence [k] in order of increasing prepended digits 9's. Each value < n in [d^^n][k] must be composite. ```

```
\\ This is the PARI/GP program code for WONplate 224 by PDG [June 2, 2023]
\\ ie. making sequence [k] in order of increasing prepended digits d's.
\\ Each value < n in [d^^n][k] must be composite.
{
d=7;
for(a=1, 100,
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;
);
\\ print(f); print(a,"  ",b);
for(i=1, a-1,
if(ispseudoprime(fd), flag++; break(); );
);
b+=2;
);
print("[",d,"^^",a,"][",b-2,"]");
b=1;
)
}

```

A000224 Prime Curios! Prime Puzzle
Wikipedia 224 Le nombre 224
```

```