diff 2bdbbf5..b1f9bc4 poisson_dirichlet_prime_predecessors_v2.txt
diff --git a/poisson_dirichlet_prime_predecessors_v2.txt b/poisson_dirichlet_prime_predecessors_v2.txt
index 7cd4192..70b6157 100644
--- a/poisson_dirichlet_prime_predecessors_v2.txt
+++ b/poisson_dirichlet_prime_predecessors_v2.txt
@@ -230,175 +230,6 @@ any special way.
This theorem strengthens several earlier results and resolves a
conjecture.
-PART II. WHAT WAS KNOWN, AND WHY IT IS HARD
-
-----------------------------------------------------------------------
-
-Chapter 5. A SHORT HISTORY
-
-For ordinary integers (not primes), the story starts in the 1930s.
-
- 5.1 Dickman (1930) and the largest prime factor of n
-
-Dickman studied: for a uniformly chosen integer n <= x, what is the
-distribution of the largest prime factor P+(n)?
-
-Write P+(n) <= x^{1/u} for a parameter u >= 1. Dickman showed:
-
- #{1 <= n <= x : P+(n) <= x^{1/u}} ~ x * rho(u),
-
-where rho is the "Dickman function:"
-
- rho(u) = 1 for 0 <= u <= 1
- u rho'(u) + rho(u-1) = 0 for u > 1.
-
-For example, rho(2) = 1 - log 2 ~ 0.307. So roughly 31 percent of
-integers up to x have largest prime factor at most sqrt(x).
-
-Dickman's work started a large field called "smooth number theory."
-N. G. de Bruijn (1951) gave the modern asymptotic form of rho(u)
-that we still use today.
-
- 5.2 Billingsley (1972): the joint law
-
-Billingsley asked about the joint distribution of the ordered prime
-factors of a random integer. He proved: as x -> infinity, the
-decreasing sequence of logarithms of large prime divisors of a
-random integer n <= x, rearranged and normalized by log x, converges
-in distribution to a certain limit. That limit is precisely the
-Poisson-Dirichlet distribution.
-
- 5.3 Donnelly and Grimmett (1993): the clean version
-
-Donnelly and Grimmett gave a "size-biased" treatment of the
-problem, which handles multiplicities cleanly and leads directly to
-the stick-breaking construction of PD(1). This is the version we use.
-
- 5.4 Arratia, Kochman and Miller (2014): factorial intensities
-
-Arratia, Kochman and Miller recast the result in terms of "joint
-intensities" of factor tuples, and gave a factorial-moment form
-that is the cleanest analytic version.
-
-Combined, these results for ordinary integers form a complete proof
-that for a uniformly chosen n <= x, the shape of the prime
-factorization of n converges to PD(1).
-
- 5.5 Ford, Konyagin and Luca (2010): the conjecture for primes
-
-The problem for primes is different. Ford, Konyagin and Luca, in
-their 2010 paper "Prime chains and Pratt trees," formulated the
-conjecture:
-
- For a uniformly chosen prime p <= x, the shape of the prime
- factorization of p - 1 also converges to PD(1).
-
-Exactly the same answer as for ordinary integers, but now on
-primes. FKL knew this was much harder than the ordinary-integer
-case: for ordinary integers, subtracting 1 is not a thing (you
-just factor n), but for primes, subtracting 1 destroys the
-structure of the number in a way that is hard to control.
-
- 5.6 Granville (2008): the smoothness conjecture for primes
-
-Granville, in his survey "Smooth numbers: computational number
-theory and beyond," stated the conjectured asymptotic:
-
- #{p <= x : P+(p-1) <= x^{1/u}} ~ pi(x) * rho(u) for u >= 1.
-
-This is the "shifted-prime Dickman conjecture." It is a
-consequence of the full PD(1) theorem.
-
- 5.7 Partial results before 2026
-
-Some partial results were known.
-
- - Baker and Harman (1998) proved lower bounds of the form
- x / (log x)^C for primes p <= x with p - 1 having no large
- prime factors. Threshold x^{0.2961}.
-
- - Lichtman (2022) improved this to threshold x^{0.2844} for
- primes in (x, 2x].
-
- - Ford (2025) and Gorodetsky (2026) proved strong results about
- small prime divisors of shifted primes, using sieve methods.
-
- - Bharadwaj and Rodgers (2026) proved PD(1) for shifted primes
- under the Elliott-Halberstam conjecture (a very strong but
- unproven distributional hypothesis about primes in arithmetic
- progressions). Unconditionally, they got "level 1/2": the
- moment formulas hold on the part of the simplex where the
- coordinates sum to less than 1/2.
-
- 5.8 2026: the full unconditional proof
-
-
-
-
-Chapter 6. WHY IS THIS HARD FOR PRIMES?
-
-For ordinary integers, Dickman, Billingsley, Donnelly-Grimmett and
-Arratia-Kochman-Miller all worked. For primes, why should it be any
-harder?
-
-The reason is subtle but important. Let's see what happens if we try
-to naively adapt the ordinary-integer proof.
-
- Counting trick for integers.
-
-Suppose we want to count integers n <= x whose prime factors include
-a specific set of "small" primes q_1, ..., q_r. That is easy: the
-answer is just floor(x / (q_1 q_2 ... q_r)), which is x / Q where
-Q = q_1 q_2 ... q_r.
-
- Counting trick for primes.
-
-Now suppose we want to count primes p <= x such that p - 1 is divisible
-by q_1 q_2 ... q_r = Q. That means p is congruent to 1 modulo Q. To
-count such primes, we need the prime number theorem for arithmetic
-progressions:
-
- #{p <= x : p ≡ 1 mod Q} = Li(x) / phi(Q) + error.
-
-For FIXED Q, or Q growing slowly like a power of log x, the error
-term is smaller than Li(x)/phi(Q) by any fixed power of log x.
-That is Siegel-Walfisz.
-
-But for the full PD(1) theorem, we need Q up to x^{1-eps}, and no
-unconditional prime-counting tool is sharp enough at that scale.
-The best we can do unconditionally is Bombieri-Vinogradov: on
-average over Q up to x^{1/2 - eps}, the error is small. Beyond that,
-it is conjectural (Elliott-Halberstam).
-
- The gap.
-
-Suppose we try to bound the number of primes p <= x with p - 1
-having all prime factors <= x^{1/u}. For u > 1, this is exactly
-Granville's conjecture. The naive count from PNT-APs is
-
- sum over Q | m for some m <= x, m has only small prime factors
- of (Li(x) / phi(Q)) + (error terms).
-
-If we ignore the errors, the main-term sum equals Li(x) times
-rho(u), which is exactly the Dickman/Granville answer. So the naive
-calculation WOULD give the right answer if the error terms were
-uniformly tiny.
-
-But the error term is x / (log x)^A for each individual modulus Q.
-We have to sum over many (potentially exp(some power of log x) many)
-moduli Q. The errors then ADD UP and swamp the main term. That is
-the whole problem.
-
-So we need a way to control these errors SIMULTANEOUSLY over all
-relevant moduli, using cancellation rather than PNT-APs. That is
-exactly what the FKL conjecture and Granville's conjecture ask for,
-and it is why the problem remained open for so long.
-
-
-
-
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-
PART III. THE PROOF
----------------------------------------------------------------------