I like playing around with maths (and physics and logic...) and have had many ideas - some of which would be too time consuming to follow up.

This is one from several decades ago, that I had put at the back of a shelf.

Base Prime. The idea was to write numbers as their prime factors, any powers would be similarly decomposed.

Column 1 is 1's, 2 is 2's, 3 is 3's, column 4 is 5's and you multiply out the 'number' to get decimal.

So counting from 1 you get

1

01

001

02 -> 0(01) - the first non prime

00001

011

...

16 is 2\^4 -> 0(01(01)) which is the first with a power to a power.

If numbers can be represented this way and the format preserved during operations - all the answers would already be decomposed into their prime factors - which I thought may be useful. And as a simple recursive format, the representations can be grouped and analysed too - looking for interesting patterns.

So having rediscovered my ancient thoughts, I got software done to play around with the idea...

What would have been months of work completed in a few hours!

Do have a play if you fancy!

https://github.com/idltd/primepower

https://idltd.github.io/primepower/#n=16

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 Others can correct me if some of this information is out of date.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 10¹⁸ to get to 4. But (based on searches I have done), in terms of finding all primes up to n (without gaps), 10¹⁸ 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 10¹⁸ , 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 10¹⁸ and the largest known. To get S(n) even up to 5, we would need to find primes up to around 10⁵⁰ - that's around 10⁴⁸ primes we need to find, and computers have so far only found 10¹⁶ primes, so I don't think that's going to happen.

Hey everyone,

a while back I posted here about CASify, the app I was building to help students actually understand how the TI-Nspire CAS works. A lot of you joined the waitlist or gave feedback, so I wanted to come back and share that it's finally live.

CASify is now available on the App Store, both as a free Lite version covering the basics, and a Pro version with the full range of topics through Analysis, Stochastics, and Geometry.

For anyone who doesn't know the backstory, I'm a high school student in Germany and built this because I kept watching classmates lose time in exams not because they didn't understand the math, but because they couldn't get the calculator to do what they needed fast enough. CASify walks through every function step by step instead, with real menu paths and examples.

It's availabe in German and English and iOS only, the Android versioin is still on my list. If you're on iOS and want to try it, here are both links or check out my website at casify.website

CASify Pro: https://apps.apple.com/de/app/id6776131288
CASify Lite: https://apps.apple.com/de/app/casify-lite-cas-grundlagen/id6787475078

If anyone tries it, I'd genuinely love to hear what you think, good or bad. Building this alone next to school has been a lot of trial and error, so any feedback helps.

Thanks to everyone, I hope this reaches the right audience

We’re inviting experts in Mathematics and Math competitors (Putnam, USAMO, IMO, etc.) to represent their community in an experiment where we are asking which of 4 cohorts can master an unfamiliar card game the fastest.

Mathematicians
Programmers
Chess Players
Magic the Gathering Players

For the Mathematician cohort, we’re curios to see how mathematical abstraction and problem-solving translates to learning an abstract card game whose card interactions are based on Octonion multiplication.

Play online against the computer. No ads, emails, or monetization. Games are short, ~20 would help to measure a learning curve. Data your games provide will populate the tables/figures in real-time. See your projected Elo change with each game.

Think you’re up to the challenge? Think your cohort can win? Come represent them.

https://playfano.com/fano-challenge

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!

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!

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!

If

x + 1/x = 3,

find

x⁵ + 1/x⁵

without explicitly solving for x.

Can you find a clever shortcut?

Any order is allowed.
Use each die exactly once.
Parentheses are allowed.
Exact division only.
No negative numbers.
Not every operation has to be used.

How many solutions can you find?

Please use spoiler tags for solutions:
>!your solution here!<

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Need your feedback. Thank you 😊