When I use the term ninedigital in these articles I always refer to a strictly zeroless pandigital (digits from 1 to 9 each appearing just once).
Ninth Page
Topic 9.3 [ October 18, 2026 Nine- & pandigitals formed from borderprimes.
From prime \(10^{4185}-{\color{blueviolet}{4613}}\) to prime \(10^{4185}+{\color{green}{29587}}\) lies no other primenumbers. That is why these primes are borderprimes .
If you study the negative and positive displacements to this \(10\)th power you'll notice that when we concatenate (combine) them
I found only five more examples in my database. Here is an overview of all six ninedigital cases and all four pandigital cases.
exponent of negative positive Resulting \(4185\) \(-4613\) \(+29587\) \(461329587\) \(5933\) \(-12947\) \(+6853\) \(129476853\) \(11766\) \(-7629\) \(+45831\) \(762945831\) \(12548\) \(-23457\) \(+1689\) \(234571689\) \(12562\) \(-27651\) \(+8943\) \(276518943\) \(21603\) \(-4827\) \(+19563\) \(482719563\)
exponent of negative positive Resulting \(8157\) \(-40569\) \(+18273\) \(4056918273\) \(15310\) \(-28169\) \(+45037\) \(2816945037\) \(22325\) \(-40817\) \(+26953\) \(4081726953\) \(23175\) \(-20457\) \(+36189\) \(2045736189\)
Here is the Pari/gp AI script that I unleashed on the database to find these solutions:
{
L=readstr("C://Pari_Patrick/borderprp(20000-24999).txt");
for(i=1,#L,
t=select(x->#x,strsplit(L[i],"\t"));
if(#t<3,next);
u=Vec(Vecsmall(t[2])); v=Vec(Vecsmall(t[3]));
if(#u<2||#v<2||u[1]!=45||v[1]!=43,next);
if(vecmin(u[2..#u])<48||vecmax(u[2..#u])>57,next);
if(vecmin(v[2..#v])<48||vecmax(v[2..#v])>57,next);
s=Str(Strchr(u[2..#u]),Strchr(v[2..#v]));
d=Set(Vecsmall(s));
n=#s; m=#d; z=setsearch(d,48);
if(n!=m,next);
if(m==10,print(t[1],"\t",s,"\tpandigitaal"));
if(m==9&&z==0,print(t[1],"\t",s,"\tninedigitaal")));
}
Topic 9.2 [ July 5, 2026 Ninedigitals equal to the sum of four fourth powers
There are in total only five solutions.
That is according to this source Sums of 4th and 5th powers containing digits 1-9
\begin{aligned}
24^{\style{font-size:20px}4}+69^{\style{font-size:20px}4}+78^{\style{font-size:20px}4}+135^{\style{font-size:20px}4}&\mathbf{\color{blue}{\;=\;}}392164578\\
27^{\style{font-size:20px}4}+35^{\style{font-size:20px}4}+68^{\style{font-size:20px}4}+149^{\style{font-size:20px}4}&\mathbf{\color{blue}{\;=\;}}516297843\\
27^{\style{font-size:20px}4}+35^{\style{font-size:20px}4}+86^{\style{font-size:20px}4}+149^{\style{font-size:20px}4}&\mathbf{\color{blue}{\;=\;}}549617283\\
48^{\style{font-size:20px}4}+57^{\style{font-size:20px}4}+96^{\style{font-size:20px}4}+123^{\style{font-size:20px}4}&\mathbf{\color{blue}{\;=\;}}329685714\\
64^{\style{font-size:20px}4}+72^{\style{font-size:20px}4}+85^{\style{font-size:20px}4}+139^{\style{font-size:20px}4}&\mathbf{\color{blue}{\;=\;}}469152738\\
\end{aligned}
Topic 9.1 [ December 22, 2025 From Henry E. Dudeney 's book “ Amusements In Mathematics”
Dudeney just gave one example namely \(981324576\) where the nine digits are so arranged that they form "four"
What about other than four squares? How many arrangements can you find that gives "five', "three" and "two" squares?
"Now, can you put the nine digits all together so they form a "single" square number?" Dudeney asked.Topic 2.2
And now it is time to delve into the many possible variations.
The analogue challenge now with pandigitals. Provide the full listings.
Four squares→ \([4][25][81][30976]\) is \([2^2][5^2][9^2][176^2]\)
Three squares→ \([9][81][4730625]\) is \([3^2][9^2][2175^2]\)
Two squares→ \([7056][321489]\) is \([84^2][567^2]\)
One square → \([1026753849]\) is \([32043^2]~~\) and \(~~[9814072356]\) is \([99066^2]\)
[
January 10-18, 2026 ]
Alexandru Petrescu came to the rescue. He provided me a list of all the solutions with two, three, four and five factors.
And even one mixed powers solution with six factors!
Note : the factor \(1\) will only be considered with exponent \(2\).
NINEDIGITAL SOLUTIONS
TWO FACTORS 2 ][8672 ]2 ][1262 ]2 ][22772 ]2 ][24392 ]2 ][2672 ]2 ][5672 ]2 ][5912 ]2 ][6182 ]2 ][6212 ]2 ][15932 ]2 ][23822 ]2 ][27272 ]2 ][30422 ]2 ][1892 ]2 ][36782 ]2 ][59042 ]2 ][80822 ]2 ][85592 ]2 ][90242 ]TWO FACTORS 3 ][763 ]3 ][763 ]3 ][2893 ]3 ][3193 ]
TWO FACTORS 7 ][6162 ]2 ][353 ]2 ][383 ]2 ][145 ]3 ][1862 ]3 ][2132 ]2 ][353 ]2 ][383 ][196][537824] = [142 ][145 ] 7 ][1862 ]7 ][2132 ]5 ][8672 ]3 ][11642 ]3 ][18662 ]3 ][18962 ]3 ][23222 ]3 ][25292 ]5 ][21262 ]2 ][385 ]9 ][763 ]3 ][2192 ]6 ][6212 ]3 ][6212 ]2 ][243 ]4 ][15932 ]3 ][37342 ]4 ][23822 ]4 ][27272 ]4 ][30422 ]3 ][46342 ]3 ][46632 ]3 ][48142 ]3 ][176 ]3 ][49132 ]3 ][72372 ]3 ][73632 ]3 ][86232 ]2 ][145 ]
THREE FACTORS 2 ][52 ][7632 ]2 ][162 ][2722 ]2 ][22 ][18162 ]2 ][22 ][19642 ]2 ][22 ][24142 ]2 ][22 ][25272 ]2 ][22 ][29272 ]2 ][22 ][30942 ]2 ][72 ][9342 ]2 ][242 ][2932 ]2 ][52 ][2722 ]2 ][282 ][732 ]2 ][82 ][7332 ]2 ][252 ][2722 ]2 ][282 ][372 ]2 ][282 ][442 ]2 ][292 ][862 ]2 ][232 ][282 ]2 ][272 ][722 ]2 ][92 ][2732 ]2 ][92 ][2822 ]2 ][42 ][7332 ]2 ][52 ][3742 ]2 ][52 ][6262 ]2 ][172 ][1242 ]2 ][182 ][872 ]2 ][242 ][572 ]2 ][272 ][662 ]2 ][182 ][1262 ]
THREE FACTORS 2 ][54 ][2722 ]4 ][52 ][2722 ]2 ][26 ][7332 ]2 ][43 ][7332 ]4 ][282 ][732 ]2 ][23 ][15932 ]2 ][23 ][23822 ]2 ][23 ][27272 ]2 ][23 ][30422 ]2 ][32 ][2033 ]3 ][62 ][2282 ]3 ][62 ][3092 ]3 ][34 ][1862 ]3 ][34 ][2132 ]5 ][72 ][1262 ]5 ][242 ][432 ]5 ][34 ][2142 ]5 ][292 ][872 ]2 ][72 ][57 ]2 ][36 ][722 ]2 ][93 ][722 ]2 ][34 ][2732 ]2 ][34 ][2822 ]2 ][24 ][7332 ]2 ][25 ][8672 ]2 ][612 ][193 ]4 ][35 ][872 ]2 ][35 ][872 ]4 ][182 ][872 ]4 ][242 ][572 ]4 ][36 ][662 ]2 ][36 ][662 ]4 ][93 ][662 ]2 ][93 ][662 ]4 ][272 ][662 ]3 ][422 ][732 ]3 ][133 ][662 ]3 ][35 ][1342 ]3 ][33 ][4422 ]3 ][33 ][5932 ]3 ][33 ][6292 ]3 ][33 ][7372 ]3 ][182 ][1342 ]3 ][192 ][2732 ]3 ][192 ][2822 ]3 ][662 ][892 ]3 ][692 ][732 ]3 ][242 ][1182 ]3 ][36 ][1162 ]3 ][93 ][1162 ]3 ][272 ][1162 ]2 ][24 ][145 ]2 ][42 ][145 ]2 ][123 ][662 ]2 ][422 ][183 ]2 ][37 ][662 ]2 ][35 ][1262 ]2 ][33 ][6212 ]2 ][25 ][8712 ]
FOUR FACTORS 2 ][62 ][232 ][282 ]2 ][22 ][32 ][9342 ]2 ][52 ][92 ][862 ]2 ][52 ][192 ][282 ]2 ][3^4][182 ][242 ]FOUR FACTORS 2 ][23 ][35 ][872 ]2 ][23 ][33 ][1862 ]2 ][23 ][33 ][2132 ]2 ][23 ][25 ][2142 ]2 ][23 ][182 ][872 ]2 ][23 ][62 ][2732 ]2 ][23 ][62 ][2822 ]2 ][23 ][242 ][572 ]2 ][23 ][36 ][662 ]2 ][23 ][93 ][662 ]2 ][23 ][272 ][662 ]2 ][32 ][62 ][353 ]2 ][32 ][62 ][383 ]5 ][72 ][34 ][242 ]5 ][72 ][92 ][242 ]2 ][52 ][34 ][862 ]2 ][25 ][34 ][872 ]2 ][25 ][92 ][872 ]3 ][32 ][33 ][1162 ]2 ][23 ][53 ][862 ]2 ][23 ][25 ][1342 ]2 ][23 ][29 ][862 ]2 ][23 ][83 ][862 ]2 ][32 ][25 ][1262 ]2 ][32 ][62 ][57 ]3 ][32 ][26 ][1892 ]3 ][32 ][43 ][1892 ]3 ][32 ][82 ][1892 ]2 ][33 ][62 ][722 ]2 ][33 ][34 ][662 ]2 ][33 ][92 ][662 ]2 ][25 ][242 ][292 ]2 ][25 ][34 ][742 ]2 ][25 ][92 ][742 ]2 ][62 ][53 ][282 ]2 ][62 ][29 ][282 ]2 ][34 ][35 ][242 ]2 ][92 ][35 ][242 ]2 ][34 ][182 ][242 ]
FIVE FACTORS 2 ][32 ][52 ][62 ][282 ]FIVE FACTORS 2 ][22 ][23 ][52 ][862 ]2 ][22 ][23 ][25 ][872 ]2 ][23 ][25 ][72 ][242 ]2 ][23 ][32 ][35 ][242 ]2 ][23 ][32 ][33 ][662 ]2 ][23 ][32 ][25 ][742 ]2 ][23 ][32 ][182 ][242 ]2 ][32 ][25 ][34 ][242 ]2 ][32 ][25 ][92 ][242 ]SIX FACTORS [1][4][8][9][32][576] = [12 ][22 ][23 ][32 ][25 ][242 ]
PANDIGITAL SOLUTIONS
TWO FACTORS 2 ][1532 ]2 ][1982 ]2 ][9452 ]2 ][2282 ]2 ][3092 ]2 ][8672 ]2 ][2282 ]2 ][3092 ]2 ][25532 ]2 ][29312 ]2 ][7592 ]2 ][44552 ]2 ][75572 ]2 ][84572 ]2 ][89732 ]2 ][97052 ]2 ][2192 ]2 ][9452 ]2 ][4562 ]2 ][7772 ]2 ][10172 ]2 ][11972 ]2 ][14282 ]2 ][17012 ]2 ][19532 ]2 ][30182 ]2 ][2672 ]2 ][5672 ]2 ][5912 ]2 ][6182 ]2 ][17462 ]2 ][21992 ]2 ][23042 ]2 ][6392 ]2 ][6512 ]2 ][57422 ]2 ][58532 ]2 ][67262 ]2 ][75482 ]2 ][83942 ]2 ][85982 ]2 ][86252 ]2 ][8552 ]2 ][101282 ]2 ][102782 ]2 ][125822 ]2 ][132782 ]2 ][134342 ]2 ][135452 ]2 ][136982 ]2 ][144422 ]2 ][147662 ]2 ][168542 ]2 ][175292 ]2 ][177782 ]2 ][207542 ]2 ][217442 ]2 ][218012 ]2 ][236822 ]2 ][238892 ]2 ][240092 ]2 ][271052 ]2 ][279842 ]2 ][287312 ]2 ][292082 ]TWO FACTORS 3 ][933 ]
TWO FACTORS 3 ][444 ]3 ][19362 ]3 ][25282 ]7 ][19762 ]2 ][933 ]4 ][933 ]3 ][5972 ]2 ][693 ]4 ][2282 ]4 ][3092 ]7 ][5972 ]5 ][25532 ]5 ][29312 ]3 ][2242 ]3 ][2262 ]3 ][552962 ]3 ][64472 ]3 ][79412 ]3 ][79472 ]2 ][933 ]5 ][8422 ]5 ][40972 ]5 ][75492 ]5 ][76212 ]5 ][76792 ]2 ][353 ]2 ][383 ][4][536870912] = [22 ][229 ] [4761][328509] = [692 ][693 ] 9 ][19362 ]9 ][444 ]9 ][25282 ][5832][104976] = [183 ][184 ] 3 ][3242 ]3 ][8432 ]6 ][17462 ]3 ][17462 ]6 ][21992 ]3 ][21992 ]6 ][23042 ]3 ][23042 ]6 ][484 ]3 ][484 ]2 ][484 ]3 ][101242 ]3 ][102142 ]4 ][57422 ]4 ][58532 ]4 ][67262 ]4 ][75482 ]4 ][83942 ]4 ][85982 ]4 ][86252 ]13 ][5982 ]3 ][147432 ]3 ][153532 ]3 ][172522 ]3 ][200892 ]3 ][214392 ]3 ][221752 ]3 ][224562 ]3 ][231132 ]3 ][263512 ]3 ][281712 ]2 ][933 ]
THREE FACTORS 2 ][52 ][19362 ]2 ][52 ][25282 ]2 ][172 ][5982 ]2 ][22 ][83052 ]2 ][22 ][92232 ]2 ][22 ][92952 ]2 ][72 ][26842 ]2 ][22 ][99162 ]2 ][72 ][29452 ]2 ][232 ][8722 ]2 ][282 ][3052 ]2 ][282 ][552 ]2 ][282 ][6252 ]2 ][282 ][7972 ]2 ][282 ][552 ]2 ][192 ][2172 ]2 ][282 ][1992 ]2 ][292 ][1762 ]2 ][242 ][332 ]2 ][242 ][992 ]2 ][282 ][952 ]2 ][92 ][5072 ]2 ][92 ][7052 ]2 ][92 ][8522 ]2 ][372 ][2952 ]2 ][442 ][2952 ]2 ][52 ][19492 ]2 ][532 ][1242 ]2 ][172 ][4162 ]2 ][552 ][1332 ]2 ][552 ][1372 ]2 ][552 ][2812 ]2 ][552 ][2862 ]2 ][732 ][2592 ]2 ][192 ][2952 ]2 ][312 ][552 ]2 ][482 ][872 ]2 ][572 ][842 ]2 ][242 ][1532 ]2 ][242 ][1982 ]2 ][272 ][2522 ]2 ][422 ][1952 ]2 ][482 ][1262 ]2 ][182 ][7592 ]2 ][62 ][28022 ]2 ][692 ][1952 ]2 ][722 ][1442 ]2 ][722 ][1742 ]2 ][242 ][6392 ]2 ][242 ][6512 ]2 ][782 ][1892 ]2 ][92 ][21752 ]2 ][92 ][25952 ]
THREE FACTORS 2 ][52 ][444 ]4 ][282 ][3052 ]2 ][842 ][293 ]2 ][282 ][58 ]2 ][282 ][254 ]2 ][23 ][57422 ]2 ][23 ][58532 ]2 ][23 ][67262 ]2 ][23 ][75482 ]2 ][23 ][83942 ]2 ][23 ][85982 ]2 ][23 ][86252 ]2 ][32 ][3023 ]5 ][242 ][332 ]5 ][242 ][992 ]2 ][26 ][733 ]2 ][43 ][733 ]2 ][82 ][733 ]3 ][332 ][662 ]3 ][62 ][7172 ]3 ][62 ][7412 ]3 ][62 ][9032 ]3 ][662 ][992 ]3 ][34 ][5972 ]5 ][332 ][742 ]5 ][432 ][842 ]5 ][72 ][7592 ]5 ][742 ][992 ]5 ][242 ][2992 ]5 ][34 ][6372 ]5 ][92 ][6372 ]5 ][34 ][6862 ]5 ][92 ][6862 ]2 ][53 ][2172 ]2 ][29 ][2172 ]2 ][83 ][2172 ]2 ][34 ][5072 ]2 ][34 ][7052 ]2 ][34 ][843 ]2 ][92 ][843 ]2 ][34 ][8522 ]2 ][28 ][733 ]2 ][44 ][733 ]2 ][162 ][733 ]2 ][25 ][25532 ]2 ][25 ][29312 ]2 ][732 ][75 ]2 ][54 ][733 ]2 ][252 ][733 ]6 ][34 ][593 ]6 ][92 ][593 ]2 ][34 ][593 ]2 ][92 ][593 ]4 ][482 ][872 ]3 ][422 ][3052 ]4 ][572 ][842 ]4 ][242 ][1532 ]4 ][242 ][1982 ]4 ][36 ][2522 ]4 ][93 ][2522 ]4 ][272 ][2522 ]2 ][36 ][2522 ]2 ][93 ][2522 ]3 ][63 ][7572 ]3 ][133 ][2522 ]3 ][482 ][1342 ]3 ][35 ][3972 ]3 ][35 ][4132 ]3 ][552 ][2192 ]3 ][182 ][3972 ]3 ][182 ][4132 ]3 ][572 ][2242 ]3 ][572 ][2262 ]3 ][62 ][10272 ]3 ][62 ][14052 ]3 ][192 ][5072 ]3 ][62 ][16482 ]3 ][62 ][21232 ]3 ][192 ][7052 ]3 ][62 ][23392 ]3 ][192 ][843 ]3 ][192 ][8522 ]3 ][62 ][27012 ]3 ][212 ][1892 ]3 ][692 ][3052 ]3 ][242 ][3232 ]3 ][242 ][3522 ]3 ][242 ][5572 ]3 ][26 ][13232 ]3 ][82 ][13232 ]3 ][842 ][1182 ]3 ][36 ][5842 ]3 ][872 ][1522 ]3 ][872 ][1792 ]3 ][892 ][2522 ]3 ][32 ][36982 ]3 ][32 ][41752 ]3 ][32 ][46162 ]3 ][32 ][59682 ]3 ][32 ][65962 ]3 ][32 ][75322 ]3 ][982 ][1892 ]2 ][53 ][8722 ]2 ][27 ][5982 ]2 ][243 ][2522 ]2 ][37 ][2522 ]2 ][211 ][1242 ]2 ][63 ][933 ]2 ][35 ][7592 ]2 ][33 ][17462 ]2 ][33 ][21992 ]2 ][33 ][23042 ]2 ][33 ][484 ]2 ][25 ][22 0]2 ][25 ][10282 ]2 ][25 ][10422 ]2 ][25 ][23962 ]2 ][25 ][78 ]2 ][25 ][25992 ]2 ][25 ][28012 ]2 ][712 ][215 ]2 ][29 ][8722 ]2 ][722 ][124 ]2 ][842 ][243 ]2 ][34 ][1753 ]2 ][92 ][1753 ]2 ][34 ][21752 ]2 ][34 ][25952 ]
FOUR FACTORS 2 ][52 ][62 ][2172 ]2 ][62 ][72 ][2952 ]2 ][62 ][282 ][952 ]2 ][22 ][32 ][26842 ]2 ][22 ][32 ][29452 ]2 ][32 ][52 ][8722 ]2 ][32 ][282 ][1752 ]2 ][52 ][92 ][1762 ]2 ][32 ][192 ][2952 ]2 ][42 ][282 ][552 ]2 ][92 ][182 ][842 ]2 ][92 ][242 ][482 ]FOUR FACTORS 3 ][23 ][43 ][593 ]FOUR FACTORS 2 ][22 ][33 ][7132 ]2 ][22 ][33 ][9162 ]2 ][23 ][482 ][872 ]2 ][23 ][33 ][5972 ]2 ][23 ][25 ][6372 ]2 ][23 ][25 ][6862 ]2 ][23 ][572 ][842 ]2 ][23 ][62 ][5072 ]2 ][23 ][62 ][7052 ]2 ][23 ][62 ][843 ]2 ][23 ][62 ][8522 ]2 ][23 ][242 ][1532 ]2 ][23 ][242 ][1982 ]2 ][23 ][26 ][593 ]2 ][23 ][82 ][593 ]2 ][23 ][36 ][2522 ]2 ][23 ][93 ][2522 ]2 ][23 ][272 ][2522 ]2 ][23 ][32 ][1353 ]2 ][23 ][32 ][21752 ]2 ][23 ][32 ][25952 ]5 ][72 ][34 ][842 ]5 ][72 ][92 ][842 ]2 ][52 ][34 ][1762 ]2 ][25 ][242 ][332 ]2 ][25 ][242 ][992 ]2 ][23 ][24 ][593 ]2 ][23 ][42 ][593 ]2 ][23 ][25 ][3972 ]2 ][23 ][25 ][4132 ]2 ][23 ][53 ][1762 ]2 ][23 ][29 ][1762 ]2 ][23 ][83 ][1762 ]2 ][32 ][25 ][7592 ]3 ][25 ][72 ][2242 ]3 ][25 ][72 ][2262 ]3 ][62 ][36 ][712 ]3 ][62 ][93 ][712 ]3 ][32 ][422 ][552 ]3 ][32 ][35 ][2242 ]3 ][32 ][35 ][2262 ]3 ][32 ][33 ][5842 ]3 ][32 ][552 ][692 ]3 ][32 ][182 ][2242 ]3 ][32 ][182 ][2262 ]3 ][32 ][62 ][8452 ]3 ][32 ][242 ][1522 ]3 ][32 ][242 ][1792 ]3 ][32 ][26 ][8552 ]3 ][32 ][43 ][8552 ]3 ][32 ][82 ][8552 ]2 ][24 ][282 ][552 ]2 ][33 ][34 ][2522 ]2 ][33 ][92 ][2522 ]2 ][25 ][292 ][842 ]2 ][34 ][35 ][842 ]2 ][34 ][182 ][842 ]2 ][34 ][242 ][482 ]
FIVE FACTORS 2 ][22 ][32 ][62 ][2952 ]FIVE FACTORS 2 ][22 ][23 ][52 ][1762 ]2 ][23 ][25 ][72 ][842 ]2 ][23 ][32 ][35 ][842 ]2 ][23 ][32 ][33 ][2522 ]2 ][23 ][32 ][182 ][842 ]2 ][23 ][32 ][242 ][482 ]SIX FACTORS [1][4][8][9][32][7056] = [12 ][22 ][23 ][32 ][25 ][842 ]
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The Nine Digits Page 9
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