u/Specialist_Umpire426

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.

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!