Research log — two smooth summands for every integer

Status (accessed 2026-08-17)

• Green, 100 Open Problems, Problem 59 (Dec 2025): still open. Record identified as Balog's $4/(9\sqrt e)\approx 0.2695$. Wooley obstruction via least quadratic non-residue for $p\equiv 3\pmod 4$. Link to Erdős Problem #334.
• Bloom #334 (page last edited 03 April 2026): OPEN, “cannot be resolved with a finite computation”. Zero claimed proofs. Two comments. Original sources [Er76e, p.272], [ErGr80, p.70], [Er82d, p.55]. Related OEIS A062241, A045535.
• Balog, Acta Math. Hungar. 54 (1989), 297–301, DOI 10.1007/BF01952060. Primary scan in compute/refs/balog-1989.pdf. Uses Fouvry's Theorem C.2 (distribution in APs to large moduli). Explicitly: the method cannot prove $N^\varepsilon$; $N^{1/2}$ is almost trivial.
• Hildebrand–Tenenbaum, J. Théor. Nombres Bordeaux 5 (1993), 411–484: Balog–Sárközy 1984b give $N^{2/5}$; Balog 1989 improves to $0.2695$; $N^\varepsilon$ unknown. Ternary: $\exp(c\sqrt{\log N\log\log N})$.
• Sárközy, Acta Math. Hungar. 67 (1995), 333–345: ternary $\exp\bigl((\sqrt{3/2}+\varepsilon)\sqrt{\log N\log\log N}\bigr)$. Not binary.
• Citation graph around Balog 1989 / Green #59 / Bloom #334 (Hildebrand–Tenenbaum, Baker 2009, Győry–Hajdu–Sárközy arXiv:2006.15307, Blomer–Grimmelt–Li–Rydin Myerson arXiv:2111.01601, Dimitrov–Vigneri–Attias arXiv:1912.11546): no later all-integers binary exponent located. Scoped miss, not a proof that none exists.

Do not claim the Landau–Ramanujan density. Do not claim a table of $G(y)$ as a bound.

Published record

Write $P^+(1)=1$ and

A uniform $F(n)\le n^{0.1}$ would imply $n(p)\le p^{0.1}$ for large primes $p\equiv 3\pmod 4$, beating Burgess.

What we certified

Theorem (trivial, audited). $F(n)<2\sqrt n+1$ for every integer $n\ge 2$.

Theorem (obstruction). If $m\equiv 7\pmod 8$ and $(-m/q)=1$ for every odd prime $q\le y$, then $m$ is not a sum of two positive $y$-smooth integers. In particular $G(y)\le A045535(\pi(y)-1)$, and $n(p)\le F(p)$ for primes $p\equiv 3\pmod 4$.

Theorem (pointwise). $F(131486759)=83$, via the splitting $649+131486110$ and the obstruction through prime $79$.

Theorem (finite $G$). The values of $G(y)$ for $y\in\{1,2,3,5,\ldots,79\}$ in compute/certs/g_certified.json are correct. In particular $G(73)=G(79)=131486759$. Independently generated smooth counts: $727473$ of 73-smooth and $808372$ of 79-smooth integers through that point.

Proposition (prefix, not a bound). On the stated ranges, with integer tests $F^q>n^p$:

• $F(n)\le n^{1/2}$ for $2\le n\le 10^6$ except $\{3,7,23\}$.
• $F(n)\le n^{2/5}$ for $2\le n\le 10^6$ except the sixteen integers $3,4,5,7,11,13,14,15,23,46,47,53,71,119,311,479$.
• $F(n)\le n^{1/3}$ for $2\le n\le 3\cdot 10^5$ except the seventy-six integers listed in compute/certs/f_exceptions_exact.json, last $18191$.

Each listed exception carries an explicit splitting that realises $F$. Cubes $8,27,125$ satisfy $F(n)=n^{1/3}$ and are not exceptions.

This is a computer-assisted prefix. It is not Balog–Sárközy made effective, and it is not a new exponent. Balog 1989 is not beaten.

Covering search (failed, verified)

Templates tried, each with a hole list in compute/certs/covering_search.json and covering_refined.json:

• short intervals of length $n^\alpha$ for $\alpha\in\{1/2,0.45,0.4,1/3,0.3\}$;
• the true trivial window $2\sqrt n+1$ (zero holes, as it must);
• finite smooth-shift lists on residue classes modulo $8,24,120$;
• square-adjustment $n=(m-k)^2+r_k$ with $k\le 256$;
• Balog's reduction to $n^\alpha$-rough $n$, then short intervals;
• the even-to-odd identity.

None of these is an infinite covering at any $\varepsilon<1/2$. The modulus-$8$ shift list at $\alpha=0.4$ fails in a single class, which is a near-miss only on a short prefix.

Classification (integer)

The Bambah–Chowla-style covering uses $m=\lfloor\sqrt n\rfloor$ and $r=n-m^2$. The two-point test for $F(n)\le n^{p/q}$ is $F^q\le n^p$. Floating-point $\mathtt{pow}$ is not this test: it misclassified $8,27,125$ as $n^{1/3}$ exceptions until the comparison was rewritten with $128$-bit integer powers.

Balog's hard case is $p(N)>N^\alpha$. His weight is the Dirichlet convolution $u*v*z$ supported on

The threshold $\alpha>4/(9\sqrt e)$ is where the $z$-sum $1-\log((4/9)/\alpha)$ stays positive. We did not make Fouvry effective, and we did not rerun this weight as a search: without the distribution theorem it is another finite template.

What would count as a dent, and why tonight is not one

A dent is an explicit $\varepsilon<1/2$ with a checkable construction that works for all large $n$, or a certified residue cover that implies an infinite statement, or an exponent below $4/(9\sqrt e)$. A last exception on $[2,10^6]$ is not $N_0$. A reproduction of A062241 is not a bound. The Vinogradov wall blocks any claim of $\varepsilon\le 0.15$ by a cover that would also bound $n(p)$.

2026-08-27 — URLs opened this session

• https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf Green #59, Dec 2025 text. Still open. Record still Balog $4/(9\sqrt{e})\approx 0.2695$. Wooley wall $1/(4\sqrt{e})$. Does not claim an effective $N_0$.
• https://www.erdosproblems.com/334 Cloudflare challenge; body not read from the live page.
• https://web.archive.org/web/20260824194602/https://www.erdosproblems.com/334 OPEN, cannot be resolved with a finite computation, last edited 03 April 2026, zero claimed proofs, two comments, A062241 and A045535. Balog exponent unchanged. Does not give a later binary exponent.
• https://web.archive.org/web/2026/https://www.erdosproblems.com/forum/thread/334 my99n 03 Mar 2026: $A062241(n)\le A045535(n-1)$, with a pointer to https://github.com/vzsky/OEIS-compute (raw PDF not retrieved). Woett 19 Nov 2025: ternary Sárközy bound; $n^{0.1}$ implies a quadratic non-residue $\le p^{0.1}$ for $p\equiv 3\pmod 4$; Erdős conjectured the ternary-shaped bound also in the binary problem. Not a proof.
• https://oeis.org/A062241 Smallest $n\ge 2$ not a sum of two $\mathrm{prime}(k)$-smooth positives. Prefix $3,7,23,71,311,479,1559,\ldots$ matches the certified $G(y)$ table. $a(30)$ still blank in this JSON.
• https://oeis.org/A045535 Least negative pseudosquare modulo the first $n$ odd primes. Prefix $7,23,71,311,479,\ldots$ tracks A062241 on the computed range, as already used on 17 August.
• https://jtnb.centre-mersenne.org/item/JTNB_1993__5_2_411_0/ and the PDF at https://jtnb.centre-mersenne.org/item/10.5802/jtnb.101.pdf Hildebrand–Tenenbaum survey. Additive paragraph (journal pp. 468–469): 1984b gives $N^{2/5}$; Balog 1989 gives $0.2695$; $N^\varepsilon$ unknown. Ternary much better. No later binary exponent in that paragraph.
• https://arxiv.org/abs/2006.15307 Győry–Hajdu–Sárközy. Additive decomposability of the set of smooth numbers. Not an all-integers $F(n)$ exponent.
• https://arxiv.org/abs/2111.01601 Blomer–Grimmelt–Li–Rydin Myerson v5. Prime plus two almost-prime squares. Not two smooth summands.
• https://arxiv.org/abs/1912.11546 Dimitrov–Vigneri–Attias v2. Cryptographic heuristic. No theorem for Green #59.
• https://www.impan.pl/shop/en/publication/transaction/download/product/91781 Ki–Maier–Sankaranarayanan, Acta Arith. 175.4 (2016). Introduction cites Balog $N^{0.2695}$ as the all-integers binary record and records Balog–Sárközy’s remark that exponential sums lower $2/5$ to $0.392$. Their theorems are almost-all square+smooth and prime+smooth. Does not beat Balog 1989.
• https://www.renyi.hu/~balog/ Author page (short HTML).
• https://www.renyi.hu/~balog/publist.pdf [9]–[10] = Studia 1984 I/II; [24] = Acta Math. Hungar. 54 (1989).

Failed / blocked this session: live erdosproblems.com (Cloudflare); vzsky sequences.pdf raw URL returned 14 bytes; Springer landing for Balog 1989 (DOI only, PDF not fetched).

No later all-integers binary exponent located. Scoped miss.