Part IV · Interactive

Same fact, different digits.

You asked whether it still applies in base 2, 3, 4. It does — and here's the honest reason, shown rather than asserted: a base is just handwriting for a quantity. Watch the same Goldbach pair get added up correctly in each base.

decimal: 28 = 5 + 23

The same pair — 5 + 23 = 28 — written and added in each base. Different digits, identical truth.

base 2

(decimal 5 + 23 = 28)

101+ 10111 ────── 11100 ✓

base 3

(decimal 5 + 23 = 28)

12+ 212 ───── 1001 ✓

base 4

(decimal 5 + 23 = 28)

11+ 113 ──── 130 ✓

Why it still applies

A base only changes the digits you write a number with — never the number itself. 28 is even, 5 and 23 are prime, and their sum is 28 whether you spell them in base 2, 3, 4, or ten. Primality, evenness, and addition are properties of the quantity, not the notation. So Goldbach isn't base-dependent — it's exactly the same statement in every base. Nothing to re-check.

What the base does change: the last digit

The one thing that shifts is which final digits a prime can even have — a prime bigger than the base can't end in a digit that shares a factor with the base:

base 2:…1(1 of 1 digits can end a large prime)

base 3:…1…2(2 of 2 digits can end a large prime)

base 4:…1…3(2 of 3 digits can end a large prime)

base 10:…1…3…7…9(4 of 9 digits can end a large prime)

Bottom line: changing base can't make Goldbach easier or harder — it can't even change the question. The difficulty from the write-up lives in the numbers themselves, not in how we spell them.