Attack log

2026-08-17 — pick

See notes/picks/2026-08-17-ideation-historical.md. Landau 4 chosen over the rest of Hilbert/Smale/Landau because it is still open and has a checkable finite list plus Iwaniec's P2 theorem as a nearby object.

2026-08-17 — first compute

Enumerate even n (odd n>1 makes n^2+1 even and >2). Trial-split n^2+1; record primes and P2s. Write compute/ tables and a verifier that ignores the tables and re-derives them.

2026-08-17 — certified prefix

compute/sieve_n2p1.py then compute/verify.py: n_max=200000, exactly 12391 primes n^2+1 (including 2=1^2+1). Verifier OK (extra=0 missing=0). First 8 values 2,5,17,37,101,197,257,401 match OEIS A002496. Not a proof of infinitude.

2026-08-17 — published record (do not claim a new prime)

Fetched the lists we will cite, before extending the sieve.

• Wolf, arXiv:0803.1456 (2008–2010): complete hunt for primes $q=m^2+1<10^{20}$. Table I: $\pi_q(10^{12})=54110$, $\pi_q(10^{14})=456362$. Constant $$C_q=\prod_{p\ge 3}\Bigl(1-\frac{(-1)^{(p-1)/2}}{p-1}\Bigr)=1.372813462818246\ldots$$ (OEIS A199401; Hardy–Littlewood conjecture E / Landau–Shanks). The better count is $$\pi_q(x)\sim\frac{C_q}{2}\int_2^x\frac{du}{\sqrt{u}\,\log u}=C_q\bigl(\mathrm{li}(\sqrt{x})-\mathrm{li}(\sqrt{2})\bigr).$$ Equivalently Bateman–Horn for $f(n)=n^2+1$: $C_{\mathrm{BH}}=C_q$ and $\#\{n\le N:n^2+1\text{ prime}\}\sim C_q\int_2^N dt/\log(t^2+1)$.
• OEIS A083844 = Wolf Table I through $10^{16}$, then Gerbicz / Wolf–Gerbicz / Grantham: complete counts of such primes $<10^n$ through $n=28$ (Grantham–Graves arXiv:2502.03513 compute all such primes up to $6.25\times10^{28}$).
• OEIS A005574 / A002496 b-files: first 10000 values of $n$ and of $n^2+1$. The 10000th $n$ is 158704, already inside the first prefix.
• Iwaniec, Invent. Math. 47 (1978): infinitely many $n$ with $\Omega(n^2+1)\le 2$ (multiplicity). Lemke Oliver: “at most two prime factors counted with multiplicity.”

A new prime not on those lists would require $n\gtrsim 2.5\times10^{14}$. We will not beat that tonight. The dent is a replayable certified prefix, a complete Iwaniec-P2 table on the same range, and a Bateman–Horn / Wolf comparison. Search residue is not a lower bound.

2026-08-17 — bugs in the first sieve

1. sqrt_minus_one used $a^{(q+1)/4}$ with $a=-1$. That formula is for residues modulo a prime $\equiv 3\pmod 4$; every $q\equiv 1\pmod 4$ fell through to a linear search. Fine at the 200000 cap, fatal if we sieve with large $q$.
2. trial_omega counted distinct prime factors $\omega$, not $\Omega$. Iwaniec's $P_2$ is $\Omega\le 2$. Counterexample: $18^2+1=325=5^2\cdot 13$ has $\omega=2$ and $\Omega=3$, so it is not an Iwaniec $P_2$. The stored count 34507 is an $\omega\le 2$ count, and the $P_2$ list was never written to disk.
3. The first sieve skipped $n=1$ until the verifier caught it. Keep that discipline.

Next: residue factor-sieve with a correct $\sqrt{-1}\bmod q$, classify by $\Omega$, store factorizations, compare to Wolf / A083844 / A005574 and to $C_q\int dt/\log(t^2+1)$.

2026-08-17 — residue sieve, Ω, Wolf match

Rewrote compute/sieve_n2p1.py against compute/n2p1_lib.py. $\sqrt{-1}\bmod q$ is now $a^{(q-1)/4}$ for the first quadratic nonresidue $a$ (so $a=2$ when $q\equiv 5\pmod 8$). The sieve divides $n^2+1$ by every prime $q\equiv 1\pmod 4$, $q\le N$, on the two progressions $n\equiv\pm r\pmod q$. Leftover factors are all $>N$, hence 1, a prime, or a product of two primes.

Self-test and $N=2000$ first. Then $N=200000$: 12391 primes (same as the first prefix), $\omega\le 2$ composites = 34507 (exactly the first sieve's "P2" count), true $\Omega=2$ composites = 31953. Difference 2554 is the $18^2+1=325$ class. Bateman–Horn at $N=200000$ is 12379.26; ratio $1.00095$.

Then $N=10^6$, 37 seconds. Counts:

• primes: 54110
• $\Omega=2$ composites: 147612
• Iwaniec $P_2$ (primes plus those): 201722
• $\omega\le 2$ composites: 158367
• unsplit leftovers: 0
• $\Omega$ histogram on the 500001 live $n$ (1 and the evens) partitions completely, max $\Omega=10$

Wolf / A083844 $\pi_q(10^k)$ matches for every reachable $k=6,\ldots,12$. In particular $\pi_q(10^{12})=54110$. OEIS A005574 prefix: the 10000th $n$ is 158704, present. First P2s: $8^2+1=5\cdot 13$, $12^2+1=5\cdot 29$; $18$ is absent.

Bateman–Horn at $N=10^6$: $C_q\int=53969.85$, count/BH $=1.00260$. Wolf-li form 53970.55. OEIS A331942 predicts 53970. Truncated Euler product for $C_q$ through $2\cdot 10^6$ is 1.37281051 against the published 1.37281346.

Did not beat Wolf or Grantham. No new prime. The new object is the complete $\Omega=2$ list with factors, compute/p2_omega2.txt.

Replay: python3 compute/sieve_n2p1.py --n-max 1000000 then python3 compute/verify.py then python3 compute/plot_counts.py.

2026-08-17 — verifier OK

compute/verify.py on the $N=10^6$ lists, 203 seconds:

• primes: claimed=54110 found=54110 extra=0 missing=0 (full Miller–Rabin rescan)
• 147612 P2 rows multiply back; every factor prime
• independent trial+Pollard of every even $n^2+1$: miss=0 extra=0
• OEIS A005574 prefix, 10000 terms
• Wolf $\pi_q(10^k)$ for $k=6,\ldots,12$

The first complete-scan attempt used a 50000 trial bound and was killed after seven silent minutes. Completeness is Pollard after a 2003 trial bound, not a residue sieve.

2026-08-17 — stop

No new prime. Infinitude not claimed. The replayable objects are the two lists, the Wolf match, and the Bateman–Horn table in compute/comparison.json / figures/counts_vs_bh.png.

2026-09-04 — literature before extending

Re-opened Wolf arXiv:0803.1456v3 HTML. Table I: $\pi_q(10^{13})=156081$, $\pi_q(10^{14})=456362$. OEIS A083844 matches. Grantham–Graves arXiv:2502.03513: complete list through $6.25\times 10^{28}$. A new prime still needs $n\gtrsim 2.5\times 10^{14}$; not attempted.

Miller–Rabin: the committed witness set $(2,3,5,7,11,13,23)$ is missing 17. OEIS A014233 / Wikipedia: bases $2,3,5,7,11,13$ only cover $n<3.474\times 10^{12}$. At $N=10^6$, $n^2+1\le 10^{12}+1$ was inside that bound. At $N=10^7$, $n^2+1\le 10^{14}+1$ is not. Added $17,19,23$ (A014233 $a(9)=3.825\times 10^{18}$).

2026-09-04 — C residue sieve at $N=10^7$

compute/q2/sieve_n2p1.c is the same residue algorithm as sieve_n2p1.py. Cross-check at $N=2000$ (lists identical) and at $N=10^6$ (identical to the committed 54110 primes and 147612 $\Omega=2$ rows, including the $\Omega$ histogram). RSS 7 MB at $N=10^6$, 1.2 s.

Then $N=10^7$, 15 s, RSS 56 MB:

• primes: 456362
• $\Omega=2$ composites: 1334083
• Iwaniec $P_2$: 1790445
• $\omega\le 2$ composites: 1421501
• unsplit: 0
• $\Omega$ histogram on the 5000001 live $n$ partitions: $1{:}456362$, $2{:}1334083$, $3{:}1590647$, $4{:}1041493$, $5{:}425651$, $6{:}120548$, $7{:}25585$, $8{:}4724$, $9{:}747$, $10{:}139$, $11{:}18$, $12{:}4$

Wolf / A083844 match for $k=6,\ldots,14$, including the two new rows $\pi_q(10^{13})=156081$ and $\pi_q(10^{14})=456362$. Prefix of both lists equals the committed $N=10^6$ files. Bateman–Horn at $N=10^7$: $C_q\int=456409.53$, count/BH $=0.99990$.

2026-09-04 — independent C verifier

compute/q2/verify_n2p1.c: Miller–Rabin every even $n$, multiply-back every P2 row, trial primes $\le 2003$ plus Pollard of every even $n^2+1$. Different algorithm from the residue sieve. 53 s:

• primes claimed=456362 found=456362 extra=0 missing=0
• 1334083 P2 rows multiply back
• complete P2 scan miss=0 extra=0

2026-09-04 — Python verify.py OK

compute/verify.py --dir compute/q2, 885 s after fixing a quadratic set(found) in the extra/missing check (that construction hung the first N=10^7 attempt). Independent of the residue sieve:

• primes claimed=456362 found=456362 extra=0 missing=0
• 1334083 P2 rows multiply back
• complete trial+Pollard scan miss=0 extra=0
• OEIS A005574 prefix, 10000 terms
• Wolf $\pi_q(10^k)$ for $k=6,\ldots,14$

Dent: independently certified extension past $N=10^6$ matching Wolf $\pi_q(10^{13})=156081$ and $\pi_q(10^{14})=456362$, plus a complete $\Omega=2$ list on the same range. Did not produce a prime off Wolf/Grantham. Infinitude not claimed.

2026-09-04 — stop

The replayable objects are compute/q2/prime_n.txt, compute/q2/p2_omega2.txt, the Wolf match through $10^{14}$, and compute/q2/comparison.json. RSS 56 MB. No new prime.

2026-09-26 — literature before $N=10^8$

Re-opened Wolf arXiv:0803.1456v3 HTML Table I: $\pi_q(10^{16})=3954181$. OEIS A083844 $a(16)=3954181$. Grantham–Graves arXiv:2502.03513 Table 1 repeats that row and continues to $6.25\times 10^{28}$. Miller–Rabin bases $2,3,5,7,11,13,17,19,23$ still cover $n^2+1\le 10^{16}+1$ (A014233 $a(9)=3.825\times 10^{18}$). $10^{16}+1$ is composite, so $\#\{n\le 10^8:n^2+1\text{ prime}\}$ is the same integer as $\pi_q(10^{16})$.

2026-09-26 — segmented C sieve at $N=10^8$

compute/q3/sieve_n2p1.c is the q2 residue algorithm in strips of five million even $n$ (about 50 MB leftover plus precomputed $\sqrt{-1}\bmod q$). Identity against the committed $N=10^6$ lists in one strip and in five strips of $10^5$ even $n$. Smoke at $N=2000$: 209 primes, 432 $\Omega=2$ rows.

Then $N=10^8$, 147 s, RSS 93 MB. First strip $n\le 10^7$ reproduced the certified q2 counts (456362 primes, 1334083 $\Omega=2$). Full range:

• primes: 3954181
• $\Omega=2$ composites: 12172983
• Iwaniec $P_2$: 16127164
• $\omega\le 2$ composites: 12914417
• unsplit: 0
• $\Omega$ histogram on the 50000001 live $n$ partitions: $1{:}3954181$, $2{:}12172983$, $3{:}15562537$, $4{:}11140174$, $5{:}5081818$, $6{:}1610082$, $7{:}386099$, $8{:}75996$, $9{:}13352$, $10{:}2346$, $11{:}355$, $12{:}64$, $13{:}14$

Wolf / A083844 match for $k=6,\ldots,16$, including the new row $\pi_q(10^{16})=3954181$. Prefix of both lists equals the certified $N=10^7$ files. Bateman–Horn at $N=10^8$: $C_q\int=3955300.85$, count/BH $=0.99972$. Last prime $n=99999966$, $n^2+1=9999993200001157$, matches OEIS A083846 $a(16)$.

2026-09-26 — independent C verifier

compute/q3/verify_n2p1.c: Miller–Rabin every even $n$, multiply-back every P2 row, trial primes $\le 2003$ plus Pollard of every even $n^2+1$. Different algorithm from the residue sieve. Bitsets for membership (12.5 MB). 742 s:

• primes claimed=3954181 found=3954181 extra=0 missing=0
• 12172983 P2 rows multiply back
• complete P2 scan miss=0 extra=0

Streaming Python verify_py.py (different language): every claimed prime retested, every P2 row multiplied back, OEIS A005574 prefix, Wolf rows through $10^{16}$. Completeness of the $P_2$ census is the C trial-plus-Pollard pass.

Dent: independently certified extension past $N=10^7$ matching Wolf $\pi_q(10^{16})=3954181$, plus a complete $\Omega=2$ list on the same range. Did not produce a prime off Wolf/Grantham. Infinitude not claimed.

2026-09-26 — stop

The replayable objects are compute/q3/n2p1.json, the Wolf match through $10^{16}$, and compute/q3/run_all.sh (regenerates the gitignored lists). RSS 93 MB. No new prime.