Grafting Numbers

[SKG_Scintill]

I don't know if anyone knows about this, but there are these numbers called Grafting numbers.

Now a number that appears within the first digits of its square root is all fine and dandy, but I went to find the numbers that appear within the first fifty digits of its square root (up to 10 million).
The result quite surprised me, I was expecting a couple in the thousands, but even seemingly random numbers in the millions repeat in the early decimal places.

So what I'm thinking, there might be a regularity in how square roots (which are mostly said to be irrational numbers without repetition) actually DO repeat in base 10. It could also be a single moment at which the number appears in its decimals, but that's regularity nontheless.

Here are the numbers with square roots that contain the original number (Big list):
"Digit" is the digit at which the number repeats in the root

  Number	                                  Square Root	  Digit
       0	                                   0E-40	  1
       1	                                       1	  1
       2	1.4142135623730951454746218587388284504414	  5
       3	1.7320508075688771931766041234368458390236	  3
       5	2.2360679774997898050514777423813939094543	 19
       6	2.4494897427831778813356322643812745809555	 23
       7	2.6457513110645907161710965738166123628616	  5
       8	2.8284271247461902909492437174776569008827	  2
      15	3.8729833462074170213895740744192153215408	 33
      17	4.1231056256176605856467176636215299367905	 12
      22	4.6904157598234297310568763350602239370346	 32
      23	4.7958315233127191135054090409539639949799	  9
      27	5.1961524227066320236190222203731536865234	 10
      39	6.244997998398398308950163482222706079483	 11
      41	6.4031242374328485311707481741905212402344	 28
      42	6.4807406984078603784382721642032265663147	 28
      43	6.5574385243020003599667688831686973571777	  5
      50	7.0710678118654755053285043686628341674805	 17
      51	7.1414284285428504261972193489782512187958	 33
      58	7.61577310586390865410066908225417137146	 10
      61	7.8102496759066539766536152455955743789673	 23
      62	7.8740078740118111255696931038983166217804	 35
      73	8.544003745317530373881709238048642873764	 18
      74	8.6023252670426266774938994785770773887634	 19
      76	8.7177978870813479517209998448379337787628	 38
      77	8.7749643873921225889489505789242684841156	  2
      78	8.8317608663278477365565777290612459182739	 13
      82	9.0553851381374173001859162468463182449341	 34
      84	9.1651513899116796579846777603961527347565	 21
      90	9.4868329805051381242719799047335982322693	 26
      92	9.5916630466254382270108180819079279899597	 32
      98	9.8994949366116653521885382360778748989105	  1
      99	9.9498743710661994299471189151518046855927	  1
     122	11.0453610171872611545040854252874851226807	 36
     132	11.4891252930760572326107649132609367370605	 28
     198	14.0712472794702883049922093050554394721985	 39
     216	14.6969384566990690643706329865381121635437	 35
     228	15.0996688705414996434228669386357069015503	 22
     241	15.5241746962600242198959676898084580898285	  4
     255	15.9687194226713113920368414255790412425995	 28
     300	17.3205080756887745963013003347441554069519	 24
     410	20.2484567313165868540636438410729169845581	 28
     413	20.3224014329015751911811094032600522041321	 38
     486	22.0454076850486018201991100795567035675049	 13
     491	22.1585198061603385610851546516641974449158	 38
     499	22.3383079036886762480662582674995064735413	 30
     502	22.4053565024080789669369551120325922966003	  9
     505	22.4722050542442310927526705199852585792542	  8
     513	22.6495033058122494651343004079535603523254	 20
     554	23.537204591879639536955437506549060344696	 22
     571	23.8956062906970423398433922557160258293152	 29
     606	24.6170672501823410982524364953860640525818	 32
     653	25.5538646783612755086778633994981646537781	 36
     680	26.0768096208105966127277497434988617897034	  5
     681	26.0959767013997776530231931246817111968994	 30
     692	26.305892875931810692691215081140398979187	 18
     706	26.5706605111728464407860883511602878570557	  4
     724	26.9072480941474196924900752492249011993408	  5
     725	26.9258240356725195852050092071294784545898	 13
     764	27.6405499221705071022370248101651668548584	  2
     765	27.6586333718786612223539123078808188438416	  2
     767	27.6947648482524577673302701441571116447449	 18
     771	27.7668867538296417762921919347718358039856	 30
     776	27.856776554368238407732860650867223739624	  6
     785	28.0178514522438000255988299613818526268005	  5
     859	29.3087017795056894442495831754058599472046	 33
     867	29.444863728670913616269899648614227771759	 11
     899	29.9833287011298992297270160634070634841919	 15
     905	30.0832179129826471353226224891841411590576	 38
     995	31.54362059117501004834593913983553647995	 38
    1139	33.7490740613724113927673897705972194671631	 16
    1569	39.6106046406767191569997521582990884780884	 18
    1666	40.8166632639170998686495295260101556777954	  4
    1942	44.0681290730614065864756412338465452194214	 37
    2672	51.691391933280343096157594118267297744751	 30
    3742	61.1718889687084939055239374283701181411743	 25
    3982	63.1030902571339993301080539822578430175781	 26
    4651	68.1982404465100557899859268218278884887695	 10
    4734	68.8040696470782933147347648628056049346924	 20
    5076	71.2460525222275009582517668604850769042969	 32
    5711	75.5711585196363131444741156883537769317627	  3
    5736	75.7363849150459458314799121581017971038818	  2
    5737	75.7429864739963960573732038028538227081299	 19
    5946	77.1103105946280038551776669919490814208984	  9
    6153	78.4410606761535262876350316219031810760498	 11
    6361	79.7558775263616581696624052710831165313721	 11
    6601	81.2465383877984095306601375341415405273438	 21
    7074	84.1070746132571116504550445824861526489258	  5
    7132	84.4511693228696458390913903713226318359375	 28
    7252	85.1586754241750725213933037593960762023926	 16
    7461	86.377080293327807680725527461618185043335	 26
    7612	87.2467764447489457779738586395978927612305	 36
    7674	87.6013698523031081322187674231827259063721	 24
    8269	90.9340420304739325274567818269133567810059	 27
    8391	91.6024017152388836393583915196359157562256	 23
    8401	91.6569691840178677466610679402947425842285	 10
    8662	93.0698662296234289215135504491627216339111	  6
    8817	93.8988817824791794919292442500591278076172	  6
    9739	98.6863719061553155142973992042243480682373	 22
    9797	98.9797959181569524389487924054265022277832	  3
    9998	99.9899994999499881487281527370214462280273	  1
    9999	99.9949998749937520869934814982116222381592	  1
   14341	119.7539143410352124874407309107482433319092	  8
   17484	132.2270774085247637685824884101748466491699	 31
   20391	142.797058793239841634203912690281867980957	 22
   22249	149.1609868564833618620468769222497940063477	 29
   23740	154.0779023740912805351399583742022514343262	  9
   29556	171.9185853827328287479758728295564651489258	 29
   36480	190.9973821810131653364805970340967178344727	 20
   39330	198.3179265724609194876393303275108337402344	 23
   41197	202.9704411977271263367583742365241050720215	  8
   48109	219.3376392687766553990513784810900688171387	 28
   48279	219.7248279098199645886779762804508209228516	  6
   48284	219.7362054828470832035236526280641555786133	 10
   54541	233.5401464416771375454118242487311363220215	 19
   59516	243.9590129509463451995543437078595161437988	 32
   59638	244.2089269457609361779759638011455535888672	 24
   64977	254.9058649776422953436849638819694519042969	  8
   68878	262.4461849598885123668878804892301559448242	 21
   74505	272.9560404167674505515606142580509185791016	 15
   76187	276.0199268168876187701243907213211059570313	 15
   76394	276.3946453895226795793860219419002532958984	  2
   77152	277.7624884681155208454583771526813507080078	 26
   77327	278.0773273749587701786367688328027725219727	  5
   80223	283.2366501708420969407598022371530532836914	 25
   83548	289.0467090281430841969267930835485458374023	 29
   89151	298.5816471252042560990958008915185928344727	 28
   95820	309.5480576582576190958207007497549057006836	 20
   97302	311.9326850459887054967111907899379730224609	 34
  168945	411.0291960432980999939900357276201248168945	 38
  295382	543.4905702953824402356985956430435180664063	 10
  299755	547.4988584462985272693913429975509643554688	 27
  345564	587.8469188487764540695934556424617767333984	 24
  390568	624.9543983363905681471806019544601440429688	 13
  416016	644.9930232180810207864851690828800201416016	 38
  446683	668.3434745697753669446683488786220550537109	 20
  457763	676.5818501851789505963097326457500457763672	 35
  465100	681.9824044651005578998592682182788848876953	 10
  513793	716.7935546585223391957697458565235137939453	 34
  555376	745.2355332376469050359446555376052856445313	 26
  571122	755.7261408737956571712857112288475036621094	 24
  590915	768.7099583067725916407653130590915679931641	 29
  642089	801.3045613248435756759135983884334564208984	 36
  689044	830.086742455268904450349509716033935546875	 14
  703674	838.8527880385211119573796167969703674316406	 32
  709200	842.1401308570920036800089292228221893310547	 12
  760655	872.1553760655265250534284859895706176757813	  8
  763672	873.8832874016987943832646124064922332763672	 38
  772006	878.6387198388197248277720063924789428710938	 22
  773019	879.2149907730190534493885934352874755859375	 10
  862060	928.4718627939136013083043508231639862060547	 35
  862925	928.9375651786292564793257042765617370605469	 12
  914920	956.514505901504890061914920806884765625	 22
  953918	976.6872580309420754929305985569953918457031	 32
  970608	985.1943970608034533142927102744579315185547	  8
  997997	998.9979979959920228793635033071041107177734	  4
  999998	999.9989999995000289345625787973403930664063	  1
  999999	999.9994999998749563019373454153537750244141	  1
 1046781	1023.1231597417780676551046781241893768310547	 23
 1073209	1035.9580107320953175076283514499664306640625	  9
 1393508	1180.469398163289952208288013935089111328125	 27
 1999146	1413.9115955391271199914626777172088623046875	 18
 2120971	1456.3553824530604288156609982252120971679688	 32
 2798254	1672.7982544228100323380203917622566223144531	  4
 2837591	1684.5150637498020387283759191632270812988281	 21
 3198386	1788.4031983867619146622018888592720031738281	  7
 3325195	1823.5117219255816962686367332935333251953125	 34
 3371714	1836.2227533717143614921951666474342346191406	 10
 3818359	1954.062179153979741386137902736663818359375	 34
 4121190	2030.0714273148125812440412119030952453613281	 24
 4653968	2157.3057270586382401234004646539688110351563	 28
 4729201	2174.67261903947292012162506580352783203125	 14
 4894112	2212.2639987126308369624894112348556518554688	 23
 5052490	2247.7744548775349358038511127233505249023438	 33
 5563436	2358.69370627048556343652307987213134765625	 16
 6411472	2532.0884660690667260496411472558975219726563	 23
 6649857	2578.7316649857152697222772985696792602539063	  8
 7462279	2731.7172254829010853427462279796600341796875	 23
 7639321	2763.9321626986434239370282739400863647460938	  2
 7690429	2773.1622743719849495391827076673507690429688	 35
 7836999	2799.46405585069078369997441768646240234375	 17
 7903466	2811.31037062790346681140363216400146484375	 13
 8053139	2837.8053139706394176755566149950027465820313	  5
 8106794	2847.2432281067945041286293417215347290039063	 10
 8359375	2891.2583765550944008282385766506195068359375	 38
 8626497	2937.0898862649742113717366009950637817382813	  9
 8651392	2941.3248715502340928651392459869384765625	 20
 9570700	3093.6547965149570700305048376321792602539063	 14

In case you're questioning my method of acquiring these numbers, here's the Java code I made

public static void main(String[] args){
		public void getGraftingNumbers(){
			System.out.printf("%8s\t%45s\t%7s%n", "Number", "Square Root", "Digit");
			for(Integer x = 0; x <= 10000000; x++){
				BigDecimal bd = new BigDecimal(Math.sqrt(x)).setScale(40,BigDecimal.ROUND_HALF_UP).stripTrailingZeros();
				String sqrt = bd.toString().replace(".","");
				if(sqrt.contains(x.toString())+1){
					System.out.printf("%8d\t%45s\t%3d%n", x, bd, sqrt.indexOf(x.toString()));
				}
			}
		}
	}

I'm not a highly schooled mathematician, I actually failed high school (not because of math, but because of biology). Maybe someone more bright can figure out the regularity.

P.S: Trying to make a square root method for BigDecimal, so I can get up to 100 decimals.

-----

Edit v1: Changed the code, max digits got inaccurate, narrowed it down to 40. Changed the println into a printf to make the results in the spoiler more synoptic.
Edit v2: Got to run the program again, apparently it's "sqrt.indexOf(x.toString())+1;"

[stargroup100]

Re: Grafting Numbers

I don't think the "regularity" you're looking for is this. Personally, I feel like this concept is a little bit too arbitrary. First few digits in the decimal representation might have some sort of significance, but to extend to 50 digits means nothing to me.

To see this, consider the following: You have 32 4-digit numbers in which the number occurs in the decimal representation within in first 50 digits. The probability that a 4-digit number can be found in these first 50 digits is (47*10^46)/(10^50) = 47/10000. There are about 9000 4 digit numbers, so the expected number of numbers that satisfy what we are looking for is 9000*(47/10000) = (9/10)*47 which is about 42, which is actually LESS than what we found.

Similarly, you have 27 5-digit numbers. The expected probability here is 46/100000, which comes out to an expected value of about 41, which is once again less than the actual turnout.

Generally, for a relatively small number of digits, based on chance alone, there are about 40 expected numbers that satisfy the given criteria for all of the integers with a certain number of digits, and since actual result turns out to actually be less than this, I don't see this as anything special.

[kaiten123]

Re: Grafting Numbers

there might be a regularity in how square roots (which are mostly said to be irrational numbers without repetition) actually DO repeat in base 10.

not sure what your goal is but assuming your looking for a square root that is a repeating decimal, it just isn't going to happen. (if i misunderstood you, i apologize).

the fact that the square root of a number which isn't a perfect square is irrational can be proven. not just hinted at or shown for a few cases, but proven in full generality (it's a fun exercise imo, if its too tough, you can try proving it for primes first, or even for a specific prime (2 being the popular choice)).
And the fact that no irrational numbers are repeating decimals can be proven as well (by showing that any repeating decimal is rational).
this, however, does not mean you won't find the number repeated in it's square root (in fact, you'd expect to find it an infinite number of times) assuming the digits are random they should appear infinitely many times with a predictable regularity, though since they're not really random some patterns will happen more/less than expected.

all that said, don't be discouraged if you can't find what you're looking for, hunting for patterns like this will probably do more for you math muscles than most of what you would do in high school anyway.

[SKG_Scintill]

Re: Grafting Numbers

Finally got to calculating the amount of times a number occurs in its decimals out of sheer chance.

I used this code:

public void getExpectedResults(){
		double prob = 0;
		for(Integer x = 1; x <= amountOfResults.toString().length(); x++){
			double xProb = 1/Math.pow(10,x)*(100-x.toString().length()-1);
			if(xProb > 1){
				xProb = 1;
			}
			xProb = xProb * (Math.pow(10,x) - (Math.pow(10,x)/10));
			prob = prob + xProb;
		}
		System.out.println("Expected results: " + (int)prob);
	}

Little clarification: the if-statement is so that the probability doesn't take the chance that the numbers occurs multiple times in its decimals.
//End spoiler

Up to a million and up to 100 decimals, the OP's code gives 419 results.
This code estimates 538 results out of sheer luck.

So you were right, it's just chance. Disappointing, but true...

P.S: Yes I know it's off a bit because of the 10-1, which should be 10-0, though the difference is minimal and the conclusion is unchanged.