A checked finding

Why a prime squared usually has last two digits that add to a prime.

The observation is real — about 60% of primes have it. But the reason is a surprise: it has almost nothing to do with primes. All figures below were computed and verified.

01It's true — 60% — but not about primes

Take a prime p, square it, add the last two digits of p², and ask if that sum is prime. It happens ~60% of the time. But the exact same 60% shows up for numbers that aren't prime at all:

set of numbers	prime digit-sum
primes only	60.0%
any number ending 1, 3, 7, 9	60.0%
composites ending 1, 3, 7, 9	60.0%

the real variableThe only thing that matters is that the number ends in 1, 3, 7, or 9 — which every prime above 5 does. 21, 33, 49 (all composite) obey the rule exactly like 23, 31, 47.

02The digit-sum is forced to be odd

A number not divisible by 2 or 5 squares to one of exactly ten possible last-two-digit endings:

p² ends in…	digits add to
…01	0 + 1 = 1
…09	0 + 9 = 9
…21	2 + 1 = 3 ✔ prime
…29	2 + 9 = 11 ✔ prime
…41	4 + 1 = 5 ✔ prime
…49	4 + 9 = 13 ✔ prime
…61	6 + 1 = 7 ✔ prime
…69	6 + 9 = 15
…81	8 + 1 = 9
…89	8 + 9 = 17 ✔ prime

Every one of them shares two features:

the two locked digits

The units digit is always 1 or 9 (odd) — since 1², 9² end in 1 and 3², 7² end in 9.
The tens digit is always even (0, 2, 4, 6, 8) — a standing quirk of squares.

So the sum is always even + odd = odd. It can only be one of 1, 3, 5, 7, 9, 11, 13, 15, 17.

◆Small odd numbers are mostly prime

That's the whole trick. The sum is pinned to a small odd number between 1 and 17 — and in that little range the primes dominate:

prime sums: 3, 5, 7, 11, 13, 17 ← six non-prime sums: 1, 9, 15 ← three (and 9 lands twice)

Primes spread evenly across all ten endings (~10% each), so you hit a prime sum six times out of ten:

in one sentenceSquaring a number that ends in 1/3/7/9 forces its last two digits to add to a small odd number (1–17), and small odd numbers are prime more often than not — so ≈60% land on a prime. The “primes” in the observation are a coincidence; the cause is the arithmetic of last digits.

Same flavour as the N−1 = (k+1)p finding: a real pattern, but once you chase it down the “prime” part evaporates and something simpler is doing the work.