A $2021$ problem: $20\sim 21$ and $43\times 47$

Question

Notice that $2021$ is a concatenation of consecutive integers: $20\sim 21$
Also $2021$ is a product of consecutive primes: $43\times 47$.

What is the next number with both of these properties?
$24502451$ is close, $4943\times 4957$ but $4951$ is in between.
$2484224843$ is close, $49831\times 49853$ but $49843$ is in between.
$715353612\sim 715353607$ isn't quite there but is $845785793\times 845785799$

Are there any other numbers with the $2021$ property?

Answer 1

If $p$ is prime and $q$ is the next prime and $pq$ has $2n$ digits, the probability that $pq$ is the concatenation of consecutive integers is heuristically about $10^{-n}$. For $pq$ to have $2n$ digits (i.e. $10^{2n-1} \le pq < 10^{2n}$), $p$ must be between about $10^{n-1/2}$ and $10^n$, and there are approximately $$\frac{10^{n}}{n \ln 10} - \frac{10^{n-1/2}}{(n-1/2) \ln 10} \sim \frac{10 - \sqrt{10}}{10 \ln(10)} \frac{10^n}{n}$$ such primes. Thus we expect about $0.297/n$ examples with $2n$ digits. Since the harmonic series diverges, there should be infinitely many, but the next one could be quite large and hard to find. I've checked by brute force that there are no further examples with up to $16$ digits.

EDIT: To expand on my comment above:

The concatenation of $y$ and $y+1$, where $y+1$ has $n$ digits, is $(10^n+1) y + 1$, and thus $\equiv 1 \mod (10^n+1)$.
Given an even positive integer $d$ and positive integer $n$, you can solve the equation $x(x+d) \equiv 1 \mod (10^n+1)$, and then check in each solution that $10^{2n} > x(x+d) \ge 10^{2n-1}$ and $x$ is prime and $x+d$ is the next prime. For $n \le 36$ and $d \le 10000$ the only solutions found are $n=2$, $d=4$, $x = 43$, $x(x+d) = 2021$ and that amazing $n=36$, $d=2$, $x=891077215721081784886888257701070827$, $x(x+d)=794018604377235322848433897872605582794018604377235322848433897872605583$.

Answer 2

I found more twin prime solutions. For $n = 396$, $d = 2$ I found four; the smallest is $x =$ $${\small 434219772837481616940726338933362452273916097301492635627291443512051997247961750150908315825206335025681182252954355463934702107964728269885096262688644034853639395077615105799339919571601786192665103151170947581386505769710635828116973131647706180301379657813220432413064536727826883282252811784121256116486385454859849292857777667719016881251956101283374338871503089431823276421261037909974359}$$

For $n = 420, d= 2$ I found $x =$

$${\small797632045320122442922746848370123218495083238021539262463793196558953236606673684986633691999805212011226776035396567202878987657452660659814874176096052099397205337415190293836597252999499063444222715671072742933640380698030812583452694556061309433747142801949662038045758937332990610602377203706320829077950947557183280793831442171456664369310820656043310649088552521802135329739094023210425721483954126295715977820771}$$

I wrote a Java program to find modular square roots using known factorizations of $10^n+1$ at https://stdkmd.net/nrr/repunit/10001.htm

The $n = 396$ solution shown may not be the second smallest twin prime solution because I did not analyse $10^n+1$ with unknown factorization (eg: $323, 392$).

Note that all these solutions involve only probable primes.

UPDATE:

I analyzed all $n \le$ 500 for which complete factorizations are known, using Shank's $(ln\ 10^n)^2$ estimate for prime gaps: $2 \le d < 5.3n^2$. The following table shows the number of solutions of $n$ for each $d$.

\begin{array}{r|l|r} d & n & total \\ \hline 2 & 36 \ 396(4) \ 420 \ 468 & 7 \\ 4 & 2 \ 70 \ 150 \ 154 \ 210(27) \ 270(8) \ 298 \ 306(2) \ 330(216) \ 350(4) \ 390(10) \ 450(519) & 791 \\ 12 & 370(3) & 3 \\ 14 & 336(2) \ 396(3) & 5 \\ 54 & 72 & 1 \\ 76 & 150(2) \ 210(16) \ 238 \ 270(4) \ 306(2) \ 330(209) \ 350(3) \ 390(15) \ 450(464) & 716 \\ 82 & 288 \ 336 \ 396(5) & 7 \\ 136 & 370(2) & 2 \\ 478 & 336(2) \ 420(2) & 4 \\ 1364 & 350 \ 370 & 2 \\ \hline \end{array}

I analyzed many of the remaining known factorizations ($10^{2021}+1$ is not one of them!). I found another twin prime solution: $n=1008$ with $d=2$, $x=$

$${\small415119285335713138107859159895921470120127901771012286705147863784377859054383205862088683121228843823644091024472578295715781891723489781341595609399956375971523325157129204031439607287157565203433091945798569106366453441681028958853009984899996962841286597261664058243897757317082387302022827796188228872367615940280481644558354868664735437056336615963301573812328050683875074515986286867031968871583403569829949341102058719072290611202615565955752049854880027669507042067569064851453754109081689314779674580036777493429088593987592215948631397812098103849006767378312184473327978598863061300578414744304808368322583101196999821431867760847600988156486022394149378510713398269801057016361947970128875578649362095899180615435692074486807012156095811778896443290518016088027948808773134635025220295845062105375107159001836188571569967805721691214024809680987944490924904786048603374165125520526174101682333287056313013054613391604106858759925794745594124906616469238525225716143856670923606283395248479131507}$$

I found three large solutions: $n=1530$ with $d=4$, $n=2442$ with $d=4$, and $n=3011$ (odd!) with $d=20690$ and $x=$

$${\small88690623974342445320706035164907895149666724533149745483024674929494064077161748069463728371848588950263928708885107056062087458873420997129551966175554795722180833514256206712208756791349423181129064370039115228768196330069003307522127855902035611295636718786724552834173077606816947565351851981240290592919330234435002605544818898704383050138705647019342509295291603847356454590383853903649034831518476190799174284971745361604565835545543751726529610775367997499873686078754015523170389723175091842178101899712480880871125974133817901518538674999703972616649632379652346395327832021456141800194701647932112291489233775928062470099635623529564608650310241193184759187003057364275380404785676729319256559060638811849744657325135099995176231775532397706618363314457193916318557002601459452270771049765689157482464270159966495073998419806308753511281083712198273207669118838966447628803262451478908503097089308456131000369654881518026667709557445311154935973975062432608109650859738547758359753161811236281693386411516599337580682444733887155249147703101490725039867545594773022646082904694935305608529996539856673800894879204151359288642251946537962882362542598385355877258435885255720463281832318996714749127620565928295871807381765069313332957959393036520560765777218866425812066806566968512664298253194097198110254106398916193178093323285510247757090049514423671929016532305341758972853095343072371415631421656596311059421635680366649419960217875499119738356663626923434263846557987005717753122760707664468827714474707911315796646846223047293968656604803860055531395007086544292734839551265327070525560544860595807571049301946018895129458512514361858289027536454436450967551887360811792658089837039529387070983777359590036217218745184175803793616945894809150630864349323615934382263207217535053701872322473625965756309360763380729005686068625698261693375747392352007831638590079855262236534947866527970435414431816193335704235728803592301557775355412534721854441649050543489915852763945114633640120859359425753981309635846954784400203166716706835000886558080563709914367339817642986577630512066137497993335131126095180539966000395250657557890723647715298981683035616063901855227374869260175148423588397623462842825972640026055333967889877654729507728851982076103225242572504527429707508826564462178655965426323365276645348503046435607191738892135058428362640796305615358118855629551341911741579489817482134866791604658301235483655504562219338888352571792484327753544399478907161167075036271824602275963945345494267916881834140919492205225308097367770069805455016950009471893642408322180438063500438940054744323617310086174876710854406021502917708656972234075774384689936433268632860789883740306973585819078135116827900294591791739792869006503475384136963506082899865375050762137583035828837915859970271078112397338135202345323724701287354928723862525723289692127491314630211880612187498071770083275673021172070970359804795314571223002210253991420896839561827296295193582310969508838235157649439819871296875917}$$

UPDATE (May 28, 2021)

I attempted to find solutions bigger than $n = 3011$, but found none! I rewrote my program in C using GMP and ran it on AWS with 96 cores. The following table summarises my results for all known complete factorizations of $10^n+1$ for $n > 3011$. For each $n$ the search was up to $d$ for the given ratio $\frac{d}{n}$. Note that a ratio of $\mathbf{2.3}$ corresponds to the expected number of primes in the range $[10^n,10^n+2.3n] = 1$, a ratio of $\mathbf{96}$ corresponds to the current record prime gap (of $8350$ for an $87$ digit prime number), and a ratio of $\mathbf{6.9}$ corresponds to the biggest solution I found: $n=3011, d=20690$.

\begin{array}{r|l} n & \frac{d}{n} \\ \hline <22292 & 100 \\ 22292 & 21 \\ 22303 & 10 \\ 23592 & 53 \\ 23734 & 3 \\ 26014 & 10 \\ 32962 & 10 \\ 46957 & 3 \\ 47248 & 12 \\ 64439 & 1.0 \\ 80363 & 1.0 \\ 95594 & 1.0 \\ 103624 & 1.0 \\ 132586 & 1.0 \\ ^*180178 & 0.10 \\ ^*268207 & 0.10 \\ ^*1600787 & - \\ \hline \end{array}

$^*$For the last three values of $n$ the largest prime factor is a probable prime. I did not analyse the last value, because I estimate that a single calculation of $x^y$ mod $z$ with 1600000 digit numbers would take about 70 hours.

UPDATE (June 11, 2021)

For the largest known solution $(n=3011, d=20690)$ the primes $x$ and $x+d$ have been certified using Primo and recorded in factordb.com - search for primes with 3011 digits.

UPDATE (July 8, 2021)

I searched for all twin prime solutions $(d = 2)$ for $ n = 1008$ and found a total of 18. The largest, a new record, is $x=$

$${\small959366284693377033651389735182738458526276121193598553484439760604241337237451497452789960716688792639495153272015332792541955520436759663588757045933885829578173457125721695213851669376206161760331060025909588497751253712340327354707310139498467167235392925523839181171301134866240381669769835527505988247393867108561444781188402425283990574068760196925230932242847785476986116528280435669208099164158860477631087793039498552963323814612240444137080246220725308557204315700968442132019195862098047534857069372757341953852762143251682584867890954165891418336661720575672057626771116234845905889545915860780107445581724924411155250266226552503036267220337257336647317889646751823745301888424349108532006368071312231688002693274043002649314803790153000719816292711634980344850814644852432359001479449220906110949158858923721913836167918792291736846984222096022979367999907004169558381802601034072450079249418137015546418369167885051996435963305528417728081298205640144099322592183826280769897515421235519976901}$$

No more were found for $1008 < n \le 268207$.

Answer 3

I've found two more solutions, bringing the total to 4 solutions. I've posted code here.

$794018604377235322848433897872605582\sim794018604377235322848433897872605583 = 891077215721081784886888257701070827\times891077215721081784886888257701070829$

$2518711810848159770018909254809359591672377471484881441744436703324716\sim2518711810848159770018909254809359591672377471484881441744436703324717 = 5018676928083894672666012088036109843105301546773725102790665815794437\times5018676928083894672666012088036109843105301546773725102790665815794441$

$353879205744237011544616255111782082608671961515039134082358165448687146\sim353879205744237011544616255111782082608671961515039134082358165448687147 = 594877471202462845078583461328011525336167267541426222873827376039101347\times594877471202462845078583461328011525336167267541426222873827376039101401$

Answer 4

I would have just commented but I don't have sufficient reputation. A quick brute-force approach in bases 2 to 16 is here and shows solutions where the second part of the concatenation contains leading zeros. Hence 249950~00249951 = 4999493*4999507.