r/CasualMath

I’ve been running a script to search for prime numbers hidden inside the decimal expansions of quotients of smaller primes, and I found an interesting pair.

When you divide a = 53874151 by b = 44453401 (both prime), the first 2,000 digits after the decimal point form a 2,000-digit prime number.

I verified the 2,000-digit sequence using standard Miller-Rabin tests to confirm its primality.

For anyone interested or skeptical, you can compute the division and run your own primality checks on those values to double-check.

A comment by u/GoldenMuscleGod reminded me of something I saw a while ago. Others can correct me if some of this information is out of date.

Define S(n) to be the sum of 1/p for all primes p less than or equal to n.

S(n) diverges - indeed Euler showed that you can bound it below with an expression that grows with log(log(n+1)) (log is natural log), and we can use log(log(n)) as an estimate of S(n) as n gets large.

That's painfully slow growth. It's not until n = 5,195,977 that S(n) reaches 3. And we have to sum up using all the primes up to around 10^18 to get to 4. But (based on searches I have done), in terms of finding all primes up to n (without gaps), 10^18 or so is roughly as far as computers have taken us - worldwide, that's all we know. So using only the primes we know, S(n) is about 4.

Of course, you will point out that we do know some primes larger than 10^18 , indeed much larger due to people searching for them. But that list is very sparse:

The largest known prime is around 10^41,024,319 and the tenth largest known prime is around 10^11,887,191 . The sum of 1/p for those ten largest primes is a rounding error (basically something less than 10^-11,887,190 ). But between those two primes there must be around 10^41,000,000 primes with a sum of 1/p of around 1, so virtually all that sum is missing from our knowledge.

Indeed S(n) where n is the largest known prime will be around 18. So there's a whole 14 contributed to S(n) by all the primes between 10^18 and the largest known. To get S(n) even up to 5, we would need to find primes up to around 10^50 - that's around 10^48 primes we need to find, and computers have so far only found 10^16 primes, so I don't think that's going to happen.

So the Euclidian proof of infinite primes uses the argument that if we take a group of primes, assume one is the final prime, and multiply them together, we just have to add 1 to the product, and it would have to be either a) prime itself or b) a product of primes not in the group.

But the logic is the exact same if we multiply them together and subtract 1. Why didn't Euclid use -1 instead of +1 in his proof?

I explored the sequence OEIS A018800: the smallest prime number that starts with the decimal digits of n.

For example:

- n = 1 → 11 (1 digit appended)

- n = 2 → 2 (0 digits appended, n is already prime)

- n = 4 → 41 (1 digit appended)

- n = 20 → 2003 (2 digits appended)

I processed every n starting from 1. Here are the key results.

=== SUMMARY ===

Total n processed : 114,890,000

Average digits appended : 1.492319

Median digits appended : 2

Min / Max digits appended : 0 / 3

=== DIGITS APPENDED HISTOGRAM ===

0 digits appended : 6,566,735 ( 5.72%)

1 digits appended : 45,535,043 ( 39.63%)

2 digits appended : 62,447,215 ( 54.35%)

3 digits appended : 341,007 ( 0.30%)

4+ digits appended: 0 ( 0.00%)

So 99.7% of cases required 0, 1, or 2 digits appended. Only 0.30% needed 3 digits, and no case with 4+ digits was found.

=== LINEAR REGRESSION MODEL ===

Model Equation : y = 6.08e-10 * x + 1.457

Slope (β₁) : 6.08e-10

Intercept (β₀) : 1.457

Pearson Correlation (r) : 0.0332

Coeff. of Det. (R²) : 0.0011

The regression line is almost flat – there's no significant trend in the number of digits appended as n grows.

=== GAP ANALYSIS ===

Minimum Gap : 1

Maximum Gap : 114,774,670,442

Average Gap : 3,079,831,155

Median Gap : 708,117,756

Largest Gap Occurred at : n = 114,889,560

Gap Histogram:

2 : 990,039 ( 0.86%)

4 : 1,627,035 ( 1.42%)

6 : 1,570,231 ( 1.37%)

8 : 1,978,363 ( 1.72%)

10 : 3,280,799 ( 2.86%)

12 : 3,287,701 ( 2.86%)

14 : 1,196,823 ( 1.04%)

16 : 1,560,687 ( 1.36%)

18 : 1,626,796 ( 1.42%)

20 : 25,351 ( 0.02%)

22 : 18,242 ( 0.02%)

24 : 49,591 ( 0.04%)

26 : 23,467 ( 0.02%)

28 : 27,826 ( 0.02%)

30 : 79,906 ( 0.07%)

32 : 39,414 ( 0.03%)

34 : 41,196 ( 0.04%)

36 : 67,727 ( 0.06%)

38 : 53,094 ( 0.05%)

40 : 61,974 ( 0.05%)

>40 : 97,283,737 ( 84.68%)

Over 84% of gaps between consecutive primes in this sequence are larger than 40. The maximum gap is over 114 billion.

=== FINAL NOTES ===

- Peak search depth: 3 digits appended (at n = 16,718).

- Total n processed: 114,890,000.

- The sequence is OEIS A018800: "Smallest prime beginning with n."

I'm not a mathematician – just curious about patterns in numbers. Feedback is welcome!

A rectangular garden has an area of 96 m².

Its length is 4 metres more than its width.

A path of 1 metre wide is made inside the garden along all four sides.

What is the area of the remaining garden?

• No shortcuts. Show your mathematical thinking!

I know this is just a fun numerical coincidence and a rational approximation of e/π, but I found this neat balance and wanted to share it!

If you divide the prime number 1367 by the two most famous constants in mathematics, you get remarkably close to whole numbers, almost perfectly balanced on either side:

1367 ÷ π ≈ 435 (exactly: 435.129...) – just slightly above.

1367 ÷ e ≈ 503 (exactly: 502.891...) – just slightly below.

This creates a highly accurate approximation formula:

435 × π ≈ 503 × e (with a tiny difference of only ~0.7)

The prime 1367 acts as a beautiful, harmonious meeting point for two constants from completely different mathematical worlds!

So, I get that TREE(3) is too big a number to represent in the universe with physical things, as there are not enough physical things, nor is there enough space to put them. And I get that even by reducing that to just trying to write the number, we are still unable to represent TREE(3), as there are more digits in the number than there are places to put them, even if each digit were only to take a planck volume, as there aren't enough of those to assign to the digits.

But, given all the space in the universe, could we write down the number that represents the number of digits in TREE(3)? Or could we write down the number of digits in that number?

How many layers of reduction before we arrive at a number that we can write down?

If

x + 1/x = 3,

find

x⁵ + 1/x⁵

without explicitly solving for x.

Can you find a clever shortcut?

57933396702876429686922708791662400986348602979985188253931383511489793001457731823088325981761829221665744176794023407056559491402467891577328326763021299466843118474637852656831938521549472347971073068161679301705472685236926463387338495220571064420250677315000599457941340849496227227628926493771018264821842230370349640102573492881424317306189569467101495834601991270039918780924506495405797923762205360790652073159333382795670426041033566699342449050309786673681670483369155689567554239898879039744147333971988258061042090970476729293484513072443614795766878726325795854855394491290821167148355514749149683707585283381546153703014210442470318180511906691108325146494219343498899382918018246586609827667470329166012110874981104800415741527586280026737848182673635645872230905234515169611121042867043956727839314198728626274066655467846183343599194761590368608472578398169740111485924046986870714883894285841394964627408094161019230662749101230783008668676907211199488107523306410531772045452853957706873238466829988649822157557103503283563398281775464911904789159515900987401574678885942493907604740891878907698622679570965569483682456042918236444719794534411171907606336090534029349351300276141892529795448751826394399153216183270385737795748770508612096374765333578237973395907265484337502903901947799663388329849198045756207969590055686607678195206367273600632909417024224754750428711236917913663419215925830944035539848749163178489614227546656090790164108195741048033614368495827231281392190063051315248070192263400801315095608512139510731469732311313898995746040563433121427776071482655904346538281010668476731132415829844984600414136781404774213539507859790229205890271721600309169926806121871750008163738773911610009508609149665332579632767397078877996926581337419351834754370411008686136818501030862345505385357198060894463821342298717851567836562984344806469613768024764967372979655179066074398198246805104576134474823016488842818077041661676098399378809713894284994865370648616800689225595431967181072865363430005250840767890912164530705704936837915584856606960687347372391339254432119085932175541392954343684716695162629271229789289404752104218596977036941910521266321726821940533986384237994403780618301379099347975260122724194454275088825587044488208965690373706904056926509324696308810974331790119456438147168585552011926921912167450509941646104076818762060881903969616431646384985895944231218505620547093874241169759205450145478746112796898626711966320965057212219958567338851356631739947125096250452942497473309299907612330435197454392788637359253116308685007014249605492659524429134513344137517101872279428202285951652856354827230765931502805341696470148698002737700823078904634554776750169178259216255903968865588749827789888950172452455448248833712309835657561369233157977405579365293671943131412034109901944892819245001657496671822581274180596255340507054499934060282320458240722454209933569735940032859109934686878274110864394924463573852015338428881961843292083566034669814619612606638283615766521897504566616272305253931938372830446073384019299355320864342734019517633662346790422915951954822645137091494126100390104510987373366328615363056042137440808225973600809566845718073791616927784260557845021823094999326904373592319407516660896764388092262510369182153559285446074990941863516247226532653142198551840063631989428776799533286215464660644129411503287306838551341184739976807097763115368031748646043780549055143428297230678053738453010234949008253769355207208167999203353157524666017029803679612131824740794652592875662818479980117505768541194835524231818203552256426752730455115752280837099763237606348192867936457993970866446264015812819179994138642295108872381709181937092290392544335464025324661284746003660247161196698209062164637264114930766444473471083408200329662059064201896721165015687487728300854501780810155844837489798144309942999091774466406270065305461848242329380636274754660519867343112275861821293501112101434868225378041813836808745417606289159904294165941408692922250601127804971962342807927743390030395048263275616935165347620718001157478088456439083590834464409622781693790883289597024043982584220069224170235863458745344365684082114430362867446193601075569803650773018026700003812298460527976219100308016537538008597751565631582745643139434508332515569645426771932483266712323523039014220800000

OFFICIAL MATHEMATICAL REGISTRATION

• Number Name: Androplazma Mycell

• Hindi Sister-Name: Anantplocks (अनंतप्लॉक्स)

• Created By: Rudra Keshri (रुद्र केसरी)

• Date of Creation: 13 August 2026

• Mathematical Value: 7 × 10\^1165097 (700000 trillion⁹⁷⁰⁹¹)

• Total Number of Zeros: Exactly 1,165,097 zeros following the digit 7.

Description: This massive mega-number was officially formulated, calculated, and named by Rudra Keshri on August 13, 2026. It contains over 1.1 million zeros, securely placing it in the global database of large numbers.

https://preview.redd.it/ahgks0aezzih1.png?width=679&format=png&auto=webp&s=15c8e00be7339ef9fdb0c031a8e5172e1b124ff0

Saw on X someone saying this is impossible to calculate.

I just got an abacus to attempt to improve mental math. Is it worth it to take classes or is it better to learn online/videos. On top of this are there any other things to do (practice etc.) to get really good at mental math.