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The exploration so far

My path through the primes.

Everything tried, plainly: the tools built, the hunches tested, and an honest verdict on each — true, false, or just shallow.

01The tabs I built

  • pairings — type an even number, see every odd + odd pairing that sums to it, and knock out any pairing whose first number is a multiple of a smaller odd. Split into three tables: neither works / one is off / both work.
  • first primes — for every even number up to a max, its list of first primes (the smaller prime of each Goldbach pair). 6 → [3], 100 → [3, 11, 17, 29, 41, 47].
  • comet — scatter of every even number vs how many Goldbach pairs it has, colored by prime factors. The famous Goldbach comet, with its bands.
  • primes — every prime as an Ulam spiral (or grid), up to 10 million. The diagonal streaks appear.
  • differences — bar graphs of the prime gaps, then the gaps between gaps, then the gaps between those, stacked down.

02Things I wondered / conjectured

true — and the coolest“Do numbers with more factors get more Goldbach pairs?”
Real and famous. More distinct small prime factors (especially 3) means more pairs — being divisible by 3 roughly doubles the count. That's what splits the comet into bands. (Hardy–Littlewood.)
false — a red herring“Even numbers with even digit sums → the primes/pairs also add to even?”
About a 50/50 coin flip. Digit-sum parity isn't the cause — carries (each worth 9) scramble it.
true — but shallow“count of first-primes + the first-primes themselves is even” (88 → 4 + 5 + 17 + 29 + 41 = 96)
True for every even number bigger than 4 — but it's just because every Goldbach pair past 4 is two odd primes, and count + sum of any list of odds is always even. Only exception: 4 (it uses the even prime 2).

03Questions about the primes themselves

  • “Are the Ulam spiral diagonals still there at 10 million?” — Yes. Short diagonal streaks appear at every scale, forever. Long single diagonals get rarer as primes thin out. (The 1963 Ulam spiral — very famous.)
  • “What's the slope of the primes — exponential or linear?” — Neither. The n-th prime grows like n · ln(n), superlinear by just a log factor. The slope is about ln(n): ~8 near the 1,000th prime, ~15.5 near the millionth. Logarithmically slow — nowhere near exponential.

04Big picture

  • The real gold is the factor / comet effect — structure that's actually there.
  • Two parity ideas turned out to be simple facts in disguise or coin flips — still useful: finding what isn't the cause narrows things down.
  • The through-line: asking the right kind of questions and actually checking them against data.

Also saved as a plain-text file: MY_PATH.txt in the project root.