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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
963196475283141696v

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 NOV ] ninedigitals substrings.
Let us find all smallest solutions from 2 to 9 ninedigital substrings.

{ 2 NOV_substrings with power 2 → none found }

2 NOV_substrings with power 3 → 2 solutions

297146853 3 =
26236953487129018576392477
368571429 3 =
50068548279613994372186589

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

3 NOV_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 NOV_substrings with power 11, 12, 13, 14, 15, 16 & 17 → none found }

4 NOV_substrings with power 18 → 1 unique solution.

126593784 18 =
69730514093015917842362635285515966162390979853386533698312687378594850347845
437769987132465705319852613947799401827653080762942141372458699063296

{ 5 NOV_substrings with power 19 upto 53 → none found }

5 NOV_substrings with power 54 → 1 unique solution.

457621839 54 =
4649048423815679016881693761707994251367846137627148837809849977509112339115
0893624664041607059823169958996145557237067013400497642008728700778227180271
6619828591635274809169528631549679091382639154700948359979571366641842161148
3357873801497413641003354898905326654648841937365068268075968566615198849564
1847508401637028241230305400402087323352464727855808613518043631524098987819
7644865743219148203100219738559091235979416283395521378773569037854028732788
952004958241

{ 6 NOV_substrings with power 55 upto 83 → none found }















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