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with some Ten Digits (pandigital) exceptions
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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).


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[ July 23, 2015 ]
Finding one or more ninedigitals as a substring in the decimal expansion
of some ninedigital raised to a power p.

What can we find in ninedigitals raised to the power 2

There are a lot of them but I will concentrate on those with the highest number of ninedigital substrings.
In the case of the power 2 this maximum is with 3 substrings.
162978354 2 =

26561943872549316
26561943872549316
26561943872549316

267453981 2 =

71531631952748361
71531631952748361
71531631952748361

294137658 2 =

86516961853724964
86516961853724964
86516961853724964

418739652 2 =

175342896157081104
175342896157081104
175342896157081104

981425736 2 =

963196475283141696
963196475283141696
963196475283141696

Let us continue with minimal four ninedigital substrings. I found one with power 3.
It is a nice four in a row solution.
896134527 3 =

719647185932647507781421183
719647185932647507781421183
719647185932647507781421183
719647185932647507781421183

Now, looking for at least 5 ninedigital substrings we have to go to power 7 already.
Powers 4, 5 & 6 yield no records.
351724698 7 =

665916932344685257919368457329168452133457461919769912185472
665916932344685257919368457329168452133457461919769912185472
665916932344685257919368457329168452133457461919769912185472
665916932344685257919368457329168452133457461919769912185472
665916932344685257919368457329168452133457461919769912185472

614925783 7 =

33247494252534659817234650550975941577997092127916131038242727
33247494252534659817234650550975941577997092127916131038242727
33247494252534659817234650550975941577997092127916131038242727
33247494252534659817234650550975941577997092127916131038242727
33247494252534659817234650550975941577997092127916131038242727

Two nice stepladder_5 solutions !

Let me thicken the plot at this point and leave behind us these rather trivial overlapping solutions.
Instead let me try to hunt for strictly NON_OVERLAPPING [ further on referred as NO ] ninedigitals substrings.
Let us find all solutions from 2 to 9 ninedigital substrings.

{ 2 NO_substrings with power 2 → none found }

2 NO_substrings with power 3 → 2 solutions

297146853 3 =
26236953487129018576392477
368571429 3 =
50068548279613994372186589

{ 3 NO_substrings with power 4, 5, 6, 7, 8 & 9 → none found }

3 NO_substrings with power 10 → 2 solutions.
It is only at this power 10 that three separated ninedigital substrings appear !

271593864 10 =
2183733009396738195247169455080723624976528132747281610186189679076884521963776638976

479635182 10 =
644332915723468747863172845933720713156733737801139368957793052380596182495377326490624

{ 4 NO_substrings with power 11, 12, 13, 14, 15, 16 & 17 → none found }

4 NO_substrings with power 18 → 1 unique solution.
126593784 18 =
69730514093015917842362635285515966162390979853386533698312687378594850347845
437769987132465705319852613947799401827653080762942141372458699063296

{ 5 NO_substrings with power 19, 20, 21, 22, 23, 24, 25, 26, ... → searching... }















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