11 is a prime number, divisible by only itself and 1. If you add its digits, 1 + 1, you get 2. 11 + 2 = 13, another prime number. And 13 + (1 + 3) = 17, a third prime number. And there it ends, because 17 + (1 + 7) = 25 = 5 x 5. I call (11, 13, 17) kangaroo primes, because one jumps to another. In base 10, the record for numbers below 1,000,000 is this:

6 primes: 516493 + 28 = 516521 + 20 = 516541 + 22 = 516563 + 26 = 516589 + 34 = 516623.

In base 16, the record is this:

8 primes: 97397 = 17,C75_{[b=16]} + 32 = 97429 = 17,C95_{[b=16]} + 34 = 97463 = 17,CB7_{[b=16]} + 38 = 97501 = 17,CDD_{[b=16]} + 46 = 97547 = 17,D0B_{[b=16]} + 32 = 97579 = 17,D2B_{[b=16]} + 34 = 97613 = 17,D4D_{[b=16]} + 38 = 97651 = 17,D73_{[b=16]}.

Another kind of kangaroo prime is found not by adding the sum of digits, but by adding their product, i.e., the result of multiplying the digits (except 0). 23 + (2 x 3) = 29. 29 + (2 x 9) = 47. But 47 + (4 x 7) = 75 = 3 x 5 x 5. So (23, 29, 47) are kangaroo primes too. In base 10, the record for numbers below 1,000,000 is this:

9 primes: 524219 + 720 = 524939 + 9720 = 534659 + 16200 = 550859 + 9000 = 559859 + 81000 = 640859 + 8640 = 649499 + 69984 = 719483 + 6048 = 725531.

But what about subtraction? For a reason I don’t understand, subtracting the digit-sum doesn’t seem to create any kangaroo-primes in base 10. But 11 in base 8 is 13 = 1 x 8^1 + 3 x 8^0 and 13_{[b=8]} – (1 + 3) = 7. In base 2, this sequence appears:

1619 = 11,001,010,011_{[b=2]} – 6 = 1613 = 11,001,001,101_{[b=2]} – 6 = 1607 = 11,001,000,111_{[b=2]} – 6 = 1601 = 11,001,000,001_{[b=2]} – 4 = 1597.

However, subtracting the digit-product creates kangaroo-primes in base 10. For example, 23 – (2 x 3) = 17. The record below 1,000,000 is this (when 0 is found in the digits of a number, it is not included in the multiplication):

7 primes: 64037 – 504 = 63533 – 810 = 62723 – 504 = 62219 – 216 = 62003 – 36 = 61967 – 2268 = 59699.

Base 2 also provides examples of addition/subtraction pairs of kangaroo-primes, like this:

3 = 11_{[b=2]} + 2 = 5 = 101_{[b=2]} | 5 = 101_{[b=2]} – 2 = 3

277 = 100,010,101_{[b=2]} + 4 = 281 = 100,011,001_{[b=2]} | 281 – 4 = 277

311 = 100,110,111_{[b=2]} + 6 = 317 = 100,111,101_{[b=2]} | 317 – 6 = 311

In base 10, addition/subtraction pairs are created by the digit-product, like this:

239 + 54 = 293 | 293 – 54 = 239

563 + 90 = 653 | 653 – 90 = 563

613 + 18 = 631 | 631 – 18 = 613

2791 + 126 = 2917 | 2917 – 126 = 2791

3259 + 270 = 3529 | 3529 – 270 = 3259

5233 + 90 = 5323 | 5323 – 90 = 5233

5297 + 630 = 5927 | 5927 – 630 = 5297

6113 + 18 = 6131 | 6131 – 18 = 6113

10613 + 18 = 10631 | 10631 – 18 = 10613

12791 + 126 = 12917 | 12917 – 126 = 12791

You could call these boxing primes, like boxing kangaroos. The two primes in the pair usually have the same digits in different arrangements, but there are also pairs like these:

24527 + 560 = 25087 | 25087 – 560 = 24527

25183 + 240 = 25423 | 25423 – 240 = 25183

50849 + 1440 = 52289 | 52289 – 1440 = 50849