Erdős problem #323: live audit, exact cubic data, and the collision threshold

Access/search date: 2026-07-28 UTC.

Result in one paragraph

The mandatory stop gate did not fire: the live page is OPEN, displays 0 claimed proofs, and has no current worker. I did not solve either question uniformly. I obtained and checked three concrete pieces of progress.

1. Relative to the only live-page computation, the exact nonnegative-cube

table extends by one decimal scale: \[ \boxed{f_{3,3}(10^9)=100\,735\,175}. \] Two independent C++ algorithms give this same integer, and a third, deliberately simple Python algorithm checks the initial entries.

1. A primary-source audit recovers rigorous solved regimes for the second

question and improves the threshold in the neighboring live discussion: \[ f_{k,3}(x)\sim \frac{\Gamma(1+1/k)^3}{6\Gamma(1+3/k)}x^{3/k} \qquad(k\geq 26). \] This follows from Salberger's published 2005 collision estimate. The nearby #325 discussion mentions only the older \(k\geq33\) theorem. More generally, Salberger--Wooley prove the same type of asymptotic for \(m\geq3\) when \(k>(2m)^{4m}\).

1. The whole problem has a precise energy bottleneck. A sufficient missing

lemma for the first question is \[ E_{k,k}(B)\ll_{k,\eta}B^{k+\eta}\quad\hbox{for every }\eta>0, \] where \(E_{k,m}(B)\) counts equal sums of two ordered \(m\)-tuples of positive \(k\)th powers. The strongest general paucity theorem verified here cannot enter the critical case \(m=k\): its hypothesis becomes the impossible inequality \(k>(2k)^{4k}\).

Nothing here claims that the finite density near \(0.10\) proves positive density, or that problem #323 is closed.

Claim labels used throughout are:

• [A] elementary-rigorous;
• [B] rigorous modulo the explicitly named theorem;
• [C] plausible/structural-unverified;
• [D] computational-only.

Statements transcribed from a web page are identified as source-audit facts, not as mathematical theorems.

0. Mandatory live-page gate

I first fetched the live #323 page, its LaTeX view, and the complete discussion with the Bright Data residential-browser route. A direct datacenter request was not used as the source of truth. I also saved rendered screenshots during the audit.

Verbatim current statement

Let $1\leq m\leq k$ and $f_{k,m}(x)$ denote the number of integers $\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true that\[f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}\]for all $\epsilon>0$? Is it true that if $m<k$ then\[f_{k,m}(x) \gg x^{m/k}\]for sufficiently large $x$?

The discussion's displayed definition makes the counting convention explicit: \(1\leq n\leq x\); the representable value \(0\) is not counted.

Exact displayed state and markers

The rendered state was:

• status OPEN;
• 1 comment on this problem;
• 0 claimed proofs for this problem;
• Likes this problem Dogmachine;
• Interested in collaborating None;
• Currently working on this problem None;
• This problem looks difficult Dogmachine;
• This problem looks tractable None;
• The results on this problem could be formalisable None;
• I am working on formalising the results on this problem None.

Thus neither the claimed-proof nor current-worker stop rule applied.

Verbatim known-results text on #323

This would have significant applications to Waring's problem. Erdős and Graham describe this as 'unattackable by the methods at our disposal'. The case $k=2$ was resolved by Landau, who showed\[f_{2,2}(x) \sim \frac{cx}{\sqrt{\log x}}\]for some constant $c>0$.

For $k>2$ it is not known if $f_{k,k}(x)=o(x)$.

The one live comment

Zeraoulia Rafik, 8 June 2026, defines the same positive-\(n\), nonnegative-base \(f_{3,3}\), describes a pair-sum bitset algorithm, and reports the exact values

at \(x=10^r\), \(0\leq r\leq8\). The comment fits a conjectural density near \(0.10\) with a secondary exponent near \(0.68\), explicitly calls this numerical evidence rather than a proof, links a Zenodo package, and discloses use of ChatGPT 5.5 to predict entries up to \(10^{16}\). I treat only the stated exact range \(r\leq8\) as a claim to reproduce; predictions and fitted parameters are not inputs to this report.

All status, marker, and transcription statements in this section are [A] as direct observations of the authoritative page. The comment's finite sequence is [D] and is independently recomputed in Section 4.

1. Primary-source literature audit

Original problem source

The marker [ErGr80] is real: P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, 1980, pp. 45--46. OCR of the scan recovers the same definition and both questions. The following page also records the two-summand Erdős--Mahler theorem and asks for the analogous three-summand result. The live page remains the authoritative transcription, as required. These source-identification facts are [A].

Verified results relevant to #323

1. Two summands. Erdős and Mahler,

On the number of integers which can be represented by a binary form, J. London Math. Soc. 13 (1938), 134--139, prove a general binary-form lower bound. Its sums-of-powers specialization, also stated explicitly by Erdős--Graham, is \[ f_{k,2}(x)\gg_k x^{2/k}\qquad(k\geq3). \] Hence the second question is already true for \(m=2\). [B]

1. Three cubes. Wooley,

Sums of three cubes, II, Acta Arith. 170 (2015), 73--100, DOI 10.4064/aa170-1-6, states in Theorem 1.1 \[ f_{3,3}(x)\gg x^{0.91709477}. \] This is still short of every \(x^{1-\epsilon}\): the numerical exponent gap is \(0.08290523\). [B]

1. Older three-summand high-degree result. Browning and Heath-Brown,

Equal sums of three powers, Invent. Math. 157 (2004), 553--573, prove the required paucity for degrees \(k\geq33\). This is the paper identified in the March 2026 discussion of the closely related open problem #325. [B]

1. The sharper published threshold. Salberger,

Counting rational points on hypersurfaces of low dimension, Ann. Sci. Éc. Norm. Supér. 38 (2005), 93--115, DOI 10.1016/j.ansens.2004.10.005, gives in Corollary 4.6 a power-saving count of non-permutation solutions of \[ x_1^d+x_2^d+x_3^d=y_1^d+y_2^d+y_3^d. \] Remark 4.7 explicitly notes that its exponent is below \(3\) for \(d>25\) and concludes \(N_d(B)=6B^3+O_{d,\epsilon}(B^{3-\delta})\). [B]

1. Arbitrarily many summands in enormous degree. Salberger and Wooley,

Rational points on complete intersections of higher degree, and mean values of Weyl sums, J. London Math. Soc. 82 (2010), 317--342, DOI 10.1112/jlms/jdq027, define \(M_{d,s}(B)\) as the \(2s\)-variable equal-sum energy and \(T_s(B)\) as its permutation-diagonal part. Corollary 1.4 states, safely in the strict form printed in the author PDF, that \[ d>(2s)^{4s} \quad\Longrightarrow\quad M_{d,s}(B)-T_s(B)\ll_{d,s}B^{s-1/2}. \tag{1.1} \] Their Corollary 1.9 records the resulting represented-integer asymptotic. [B]

1. Recent cross-check. De la Bretèche and Tenenbaum,

Mean values of arithmetic functions and application to sums of powers, arXiv:2403.19320v6 (4 August 2025), Proposition 5.2, explicitly cites Salberger for the three-term threshold \(d\geq26\). It also mentions a private communication suggesting \(d\geq16\), but says that proof was to be written up. I do not use the private communication as a theorem. The published \(d\geq26\) statement is [B]; the possible improvement is [C].

I searched the exact notation and statement, “number of integers represented as sums of powers,” equal-sum/paucity papers, and recent 2024--2026 sources. I verified the theorem numbers above in primary PDFs. I found no claimed uniform solution of either question and no post-2015 unconditional exponent for \(f_{3,3}\) improving the cited Wooley value. This is an honest search result, not a proof of bibliographic completeness. [C]

2. Exact energy reduction

For integers \(k,m,B\geq1\), put

and define the ordered equal-sum energy

If

then

Cauchy--Schwarz therefore gives

All these sums are at most \(mB^k\). Taking

in (2.2) yields the exact reduction

Equations (2.1)--(2.3) are [A]. The checker recomputes the two sides on several finite examples and also computes each energy by a direct double-tuple loop.

Two consequences isolate the targets:

• If \(m<k\) and \(E_{k,m}(B)=O_{k,m}(B^m)\), then

\(f_{k,m}(x)\gg x^{m/k}\). [A]

• If, for every \(\eta>0\),

\(E_{k,k}(B)\ll_{k,\eta}B^{k+\eta}\), then, by choosing \(\eta=k\epsilon\), \[ f_{k,k}(x)\gg_{k,\epsilon}x^{1-\epsilon}. \] Thus this critical energy estimate is a sufficient lemma for the first question. [A]

The permutation-diagonal solutions already contribute

with the \(o(B^m)\) correction coming from tuples with repeated coordinates. Consequently \(B^m\) is the optimal energy scale. [A] The criterion is sufficient, not logically necessary: a different method could conceivably give a large sumset despite larger global energy.

3. Rigorous regimes obtained from the reduction

\(m=1\) and \(m=2\)

Under the positive-\(n\) convention,

This settles \(m=1<k\) directly. [A] The Erdős--Mahler theorem quoted above settles \(m=2<k\) for every \(k\geq3\). [B]

Three summands: why the exact threshold is \(k=26\)

For \(13<d\leq34\), Salberger's Corollary 4.6 bounds the non-permutation collision count by \(O_{d,\epsilon}(B^{\theta(d)+\epsilon})\), where

For \(d>34\) the exponent is \(131/45\). Direct high-precision evaluation in the checker gives

Thus every \(d\geq26\) has \(o(B^3)\) non-permutation collisions. [B]

Here is the complete passage from collisions to represented integers. The number of unordered positive triples satisfying \(a_1^d+a_2^d+a_3^d\leq x\) is, by a Riemann-sum lattice count,

The volume is the standard beta integral

Tuples with repeated coordinates number only \(O(x^{2/d})\). If \(q_n\) is the number of unordered triples producing \(n\), then

and the right side is bounded by Salberger's non-permutation collision count with \(B=\lfloor x^{1/d}\rfloor\). Hence merging distinct triples loses only \(o(x^{3/d})\) values. Allowing zero coordinates adds at most \(O(x^{2/d})\) boundary triples. Therefore

The lattice and collision-to-support deductions are [A]; the nontrivial collision bound is [B]. Equation (3.3) proves the second question for \(m=3\), \(k\geq26\).

At the last degree before the threshold, (2.3), (3.1), and \(\theta(25)=3.0084\) still give the concrete near-miss

whereas the requested exponent is \(3/25=0.12\). The exact exponent gap is \(0.000336\). [B]

Arbitrary fixed \(m\) in sufficiently high degree

Apply (1.1) with \(s=m\geq3\). Since the diagonal term is \((m!+o(1))B^m\), all but \(O(B^{m-1/2})\) tuples are essentially unique up to permutation. Repeating the simplex argument above gives

This is [B] modulo Salberger--Wooley, with the final conversion [A]. The explicit threshold is enormous:

For \(m=3\), Salberger's \(k\geq26\) result is vastly sharper.

These are partial regimes, not a uniform answer. In particular, the verified chain above does not settle \(m=3,\ 4\leq k\leq25\), nor the broad moderate-degree range for \(m\geq4\).

4. Exact computation of \(f_{3,3}(10^r)\)

The standalone checker recomputed the following table. Every row is [D].

The first nine rows reproduce the live comment; the final row extends that displayed range. I claim novelty only relative to the live #323 material, not to every database or unpublished computation.

Algorithm A: unordered triples

For \(X=10^9\), every base lies in \(0\leq a\leq1000\). The first algorithm enumerates exactly the triples

and sets the corresponding bit in a fresh \(X+1\)-bit bitmap. Ordering the bases loses no represented value. It visited exactly

admissible unordered triples and obtained \(100\,735\,175\) set positive positions. [D]

Algorithm B: distinct pair sums plus shifts

The second algorithm does not call the triple routine. It forms and sorts all values \(a^3+b^3\leq X\) with \(a\leq b\), deduplicates them, then for each cube \(c^3\) shifts every allowable distinct pair sum into another fresh bitmap. At \(X=10^9\) it found

and made

Its count at every power-of-ten checkpoint was bit-for-bit numerically equal to Algorithm A's prefix count. [D]

Third small oracle and resource use

A separate Python routine loops over all ordered triples for \(x\leq10^4\), stores their sums in a set, and reproduces the first five rows. This deliberately uses neither unordered enumeration nor pair-sum shifts. [D]

On the final full run, the two large algorithms took approximately \(3.14\) and \(4.97\) seconds. Peak resident memory was \(129\,268\) KiB; the main object is the \(10^9+1\)-bit bitmap. [D] Zero is set internally but subtracted from every reported count, matching the live convention.

The density column is only a finite table. Its proximity to \(0.10\) is compatible with positive density but does not rule out slow decay, and is not used in any proof. [C]

For comparison, James Maynard's 2026 survey Sums of three positive cubes tabulates the different positive-base convention. Those values are smaller; they should not be mistaken for a disagreement with the nonnegative-base table here.

5. The exact wall for the open cases

Why the high-degree theorem cannot touch \(m=k\)

The Salberger--Wooley hypothesis for the \(2m\)-variable energy is

In the first question \(m=k\), this becomes

which is false for every positive integer \(k\). Thus that theorem proves large-degree results only when the number of summands is tiny relative to the degree; it gives nothing on the diagonal \(m=k\). [A]

The precise sufficient missing input is:

Critical monomial paucity lemma. For a fixed \(k>2\) and every \(\eta>0\), prove \[ > \#\left\{(\mathbf a,\mathbf b)\in[1,B]^k\times[1,B]^k: > \sum_{i=1}^k a_i^k=\sum_{i=1}^k b_i^k\right\} > \ll_{k,\eta} B^{k+\eta}. > \tag{5.1} > \]

By Section 2, (5.1) would prove the first question for that \(k\). [A] The diagonal alone has order \(B^k\), so no exponent smaller than \(k\) is possible.

The modern Vinogradov mean value theorem does not supply (5.1): it counts tuples satisfying the simultaneous equations in degrees \(1,2,\ldots,k\), whereas (5.1) imposes only the single degree-\(k\) equation. Bounding the much smaller simultaneous-solution set cannot upper bound the one-equation energy. [A] Ordinary circle-method asymptotics also require more variables than the critical \(2k\) in precisely the small variable regime at issue. No verified paper in the audit bridges that gap. [C]

For \(k=3\), Wooley's \(0.91709477\) exponent leaves an explicit gap of \(0.08290523\) from \(1\). The exact \(10^9\) count cannot fill a uniform asymptotic gap: any finite table is compatible with eventual decay below every fixed positive density. [A]

Why more of the same computation is not a proof route

Both bitset algorithms use \(X/8+O(1)\) bytes and empirically near-linear work in \(X\). A run through \(10^{10}\) would require about \(1.25\) GB for each sequential bitmap and roughly \(0.02\)--\(0.05\) core-hours by linear extrapolation from this machine (well under one cent of CPU at \(\$0.05\) per core-hour, with RAM-instance cost dominating). A run through \(10^{12}\) would require about \(125\) GB and roughly \(2\)--\(5\) core-hours, normally a few dollars on a high-memory instance. I ran neither. These are engineering estimates, hence [D]. Neither finite extension could establish (5.1) or any uniform lower bound as \(x\to\infty\).

6. Reproducibility

The complete standalone checker is runs/erdos323_wave8r_reverify.py. It uses the Python standard library and a C++17 compiler. The C++ source for both independent large algorithms is embedded in the .py, compiled from standard input in a temporary directory, and deleted after the run.

Run the full check with:

python3 runs/erdos323_wave8r_reverify.py

For a lower-memory check through \(10^8\):

python3 runs/erdos323_wave8r_reverify.py --quick

The full checker independently:

1. rebuilds both \(10^9\) bitmaps;
2. asserts both methods equal the committed table at every checkpoint;
3. checks through \(10^4\) by ordered Python brute force;
4. computes several finite energies by both fibre squares and direct tuple

pairs, and checks Cauchy--Schwarz;

1. recomputes \(\theta(25)\), \(\theta(26)\), \(131/45\), the degree-25

lower-bound exponent, Wooley's exponent gap, and the explicit Salberger--Wooley thresholds.

Observed full output:

Exact f_{3,3}(10^r), with positive n only:
  x=         1  f=         1  density=1.000000000
  x=        10  f=         6  density=0.600000000
  x=       100  f=        28  density=0.280000000
  x=      1000  f=       173  density=0.173000000
  x=     10000  f=      1352  density=0.135200000
  x=    100000  f=     11662  density=0.116620000
  x=   1000000  f=    107875  density=0.107875000
  x=  10000000  f=   1037872  density=0.103787200
  x= 100000000  f=  10172774  density=0.101727740
  x=1000000000  f= 100735175  density=0.100735175
Algorithm A: unordered_triples=119249743 max_base=1000 seconds=...
Algorithm B: pair_shift_visits=355734531 distinct_pair_sums=440960 seconds=...
Ordered-triple Python brute force agrees through x=10^4.
Finite energy checks (k,m,B): support, energy, tuple_count
  (2, 2, 7): (27, 95, 49)
  (3, 2, 6): (21, 66, 36)
  (3, 3, 5): (35, 545, 125)
  (4, 3, 4): (20, 256, 64)
Salberger theta(25)=3.0084
Salberger theta(26)=2.9953500163803206241042435378762411890119423424415
First d in 14..34 with theta(d)<3 is d=26; 131/45<3.
At d=25: Cauchy exponent=0.119664, target=0.12, gap=0.000336.
Wooley cubic exponent gap to 1: 0.08290523.
Salberger--Wooley explicit thresholds: (2*3)^(4*3)=2176782336,
(2*4)^(4*4)=281474976710656.
ALL CHECKS PASSED

All tabulated and runtime statements are [D]. The code verifies the arithmetic and finite combinatorics, while the cited asymptotic theorems remain literature-dependent [B] inputs.

The verified checker source SHA-256 is 6654560a4f84cbabf6db5365da0bfb937c65104db05d5a143ae46f90116808f0.

7. Honest final state

The second question is rigorously true in several substantial regimes: \(m=1\), \(m=2\), \(m=3,\ k\geq26\), and \(m\geq3,\ k>(2m)^{4m}\). The exact cubic computation extends the live nonnegative-cube table to \(10^9\). The first question and the remaining moderate-degree second-question cases remain open under the verified literature. The sharp wall is not a missing numerical experiment but a critical one-equation energy/paucity estimate of the scale (5.1).

PARTIAL: Verified \(f_{3,3}(10^9)=100735175\) by two independent exact algorithms, derived \(f_{k,3}(x)\sim \Gamma(1+1/k)^3x^{3/k}/(6\Gamma(1+3/k))\) for every \(k\ge26\) from Salberger, and isolated the sufficient critical-energy lemma; the uniform \(m=k>2\) question remains open.