Erdős problem 325 — wave 8r

Date of live-page and literature audit: 2026-07-28 UTC.

Outcome

This run does not solve the problem for every \(k\), so the final status is

PARTIAL. It does produce three verifiable advances/corrections.

1. Published solved range corrected from \(k\ge 33\) to \(k\ge 26\).

Salberger's 2005 Corollary 4.6 and Remark 4.7 imply, for every fixed

\(k\ge 26\),

\[ f_{k,3}(x)\sim \frac{\Gamma(1+1/k)^3}{6\Gamma(1+3/k)}x^{3/k}. \]

This is (b), rigorous modulo Salberger's published theorem. The live page's

March 2026 comments mention only the weaker \(k\ge33\) threshold from

Browning--Heath-Brown.

2. A sharper published lower exponent for \(k=23,24,25\). Applying

Cauchy--Schwarz directly to Salberger's explicit non-diagonal bounds gives

\[ f_{k,3}(x)\gg_{\epsilon,k}x^{\beta_k-\epsilon}, \]

with

\[ \begin{array}{c|c|c|c} k&\beta_k&3/k&\text{Vaughan's }3/k-1/k^2\\ \hline 23&0.128818688796562&0.130434782608696&0.128544423440454\\ 24&0.124070505649870&0.125000000000000&0.123263888888889\\ 25&0.119664000000000&0.120000000000000&0.118400000000000 \end{array} \]

This is (b), rigorous modulo Salberger's theorem. It does not close these

cases.

3. Exact finite table and collision certificate. A from-scratch

standard-library program computes \(f_{k,3}(200^k)\) for every

\(3\le k\le32\), and proves that the \(1,373,701\) unordered triples

\(0\le a\le b\le c\le200\) have distinct \(k\)-th-power sums for every

\(7\le k\le32\). This is (d), computational-only. It supplies evidence and

exact finite data, not a uniform theorem.

The fully verified open range left by the published literature found here is

\(3\le k\le25\). There is an unproved 2019 announcement claiming the

asymptotic for \(k\ge11\), discussed below, but it is not used in any result in

this report.

Claim labels follow the requested scheme: (a) elementary-rigorous,

(b) rigorous modulo the named published theorem, (c)

plausible/structural-unverified, and (d) computational-only. A section's

opening label applies to its substantive claims unless an inline label

overrides it. Browser transcriptions and bibliographic existence checks are

tagged (a) because they are directly reproducible observations, not

mathematical appeals to an unproved assertion.

0. Mandatory live-page audit

Classification: (a), directly reproduced browser observations.

The page was fetched through the Bright Data browser route, not datacenter

curl.

Authoritative page:

Erdős Problem #325.

Discussion thread:

#325 discussion.

Verbatim current statement

> Let \(k\geq 3\) and \(f_{k,3}(x)\) denote the number of integers \(\leq x\)

> which are the sum of three nonnegative \(k\)th powers. Is it true that

> \[ > f_{k,3}(x)\gg x^{3/k} > \]

> or even \(\gg_\epsilon x^{3/k-\epsilon}\)?

Status and every listed marker

• Page status: OPEN.
• Comments: 2.
• Claimed proofs: 0.
• Likes this problem: None.
• Interested in collaborating: None.
• Currently working on this problem: None.
• This problem looks difficult: None.
• This problem looks tractable: None.
• Results could be formalisable: None.
• Working on formalising the results: None.
• External database: formalised statement Yes; possible related OEIS

sequences A004825, A004832, A004843.

Thus none of the mandatory stop conditions was present.

Results and comments shown on the page

The page states:

• Erdős--Mahler proved \(f_{k,2}(x)\gg x^{2/k}\).
• For \(k=3\), Wooley proved

\(f_{3,3}(x)\gg x^{0.917\ldots}\).

• The statement has been formalised in Lean in the Google DeepMind Formal

Conjectures project.

The two comments, both dated 2026-03-09, say:

1. JonathanGabor points to Browning--Heath-Brown and says its corollary proves

the statement for \(k\ge33\).

2. TerenceTao supplies the published reference: T. D. Browning and

D. R. Heath-Brown, Equal sums of three powers, Invent. Math. 157 (2004),

553--573.

The page warns that comments are user-provided and unverified. I therefore

checked the paper itself before using it.

1. Primary-source literature audit

1.1 Sources that were opened and checked

**Classification: (a) for bibliographic existence and exact transcription;

(b) whenever a mathematical conclusion invokes the named published

theorem.**

1. P. Erdős and K. Mahler,

On the Number of Integers Which Can Be Represented By a Binary Form,

J. London Math. Soc. 13 (1938), 134--139. Its theorem gives the cited

\(x^{2/k}\) lower order for the binary form \(X^k+Y^k\). The 2019

Documenta Mathematica item on the Erdős page is a reprint, not the original

publication.

2. R. C. Vaughan,

A new iterative method in Waring's problem,

Acta Math. 162 (1989), 1--71. Theorem 1.4 states, for \(k\ge4\),

\[ f_{k,3}(x)\gg_{\epsilon,k}x^{3/k-1/k^2-\epsilon}. \]

3. T. D. Wooley,

Breaking classical convexity in Waring's problem,

Invent. Math. 122 (1995), 421--451. Its corollary to Theorem 1.3 gives

\(N_{k,3}(X)\gg_k X^{3/k-e^{-k/17}}\). This is aimed at very large \(k\)

and is weaker than the relevant bounds below in the finite range at issue.

4. T. D. Wooley,

Sums of three cubes, II,

Acta Arith. 170 (2015), 73--100. Theorem 1.1 gives the explicit current

unconditional cubic exponent

\[ 0.91709477. \]

5. T. D. Browning and D. R. Heath-Brown,

Equal sums of three powers,

Invent. Math. 157 (2004), 553--573,

DOI 10.1007/s00222-004-0360-9.

The paper proves that nontrivial solutions of

\[ x_1^d+x_2^d+x_3^d=x_4^d+x_5^d+x_6^d,\qquad x_i\le B, \]

are \(o(B^3)\) for \(d\ge33\), and its corollary gives

\[ f_{d,3}(x)\sim \frac{\Gamma(1+1/d)^3}{6\Gamma(1+3/d)}x^{3/d}. \]

This validates the live comment, but it is not the sharp published

threshold.

6. P. 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.

Corollary 4.6 gives explicit upper bounds for non-permutation solutions.

Remark 4.7 observes that the relevant exponent is \(<3\) when \(d>25\)

and concludes

\[ N_d(B)=6B^3+O_{d}(B^{3-\delta_d})\qquad(d\ge26) \]

for some \(\delta_d>0\). The remark explicitly says this improves the

Browning--Heath-Brown threshold \(d>32\).

7. R. de la Bretèche and G. Tenenbaum,

Mean values of arithmetic functions and application to sums of powers,

Math. Proc. Cambridge Philos. Soc. 180 (2026), 1--13,

DOI 10.1017/S0305004125101382.

Proposition 5.2 explicitly invokes Salberger's Corollary 4.6 to record

\[ V_2(x;c,\ell)\ll x^{3/\ell} \]

for three equal exponents \(\ell\ge26\). The introduction says Salberger

privately indicated a reduction to \(\ell\ge16\), but also says that proof

remains to be written. I do not use the private claim.

8. P. Salberger,

Equal sums of three \(d\)th powers,

Oberwolfach Report 50/2019, p. 3188. The report announces

\[ N_d(B)=6B^3+O_d(B^{3-\delta})\qquad(d\ge11). \]

It gives only a short proof strategy, not a proof. No matching full paper

or preprint was found in Salberger's publication list or exact-title

searches through 2026-07-28. Since the 2025/26 de la

Bretèche--Tenenbaum paper still distinguishes the published \(26\) from a

private \(16\), I classify the \(11\) announcement as

(c) plausible/structural-unverified, and do not cite it as closing any

case.

9. J. Maynard,

Sums of three positive cubes,

J. London Math. Soc. 113 (2026), e70554. This May 2026 survey identifies

Wooley's \(0.91709477\ldots\) as the current unconditional world record for

\(k=3\), and discusses Victor Wang's conditional work.

10. V. Y. Wang,

Sums of cubes and the Ratios Conjectures,

arXiv:2108.03398v2. This is conditional on strong analytic hypotheses and

is not used as an unconditional result.

1.2 Search coverage and misses

**Classification: (c): this is a documented search result, not a proof that

no uncatalogued result exists.**

I searched exact titles, the phrases “equal sums of three powers,” “number of

integers representable as the sum of three powers,” the equation in six

variables, Salberger's publication list, and forward citations of the

Browning--Heath-Brown DOI. The forward-citation metadata returned ten records.

The only directly relevant recent paper was de la Bretèche--Tenenbaum, which

confirms the published \(26\) threshold and labels \(16\) as private

communication. I found no full proof of the announced \(11\) or private \(16\)

threshold, and no unconditional improvement on Wooley for \(k=3\).

The similarly titled 2011 Olivier Robert paper concerns mixed sums beginning

with a square, not three equal \(k\)-th powers, so it is not a result on this

problem.

2. Published theorem: the asymptotic for every \(k\ge26\)

This section is **(b), rigorous modulo Salberger (2005), Corollary 4.6 and

Remark 4.7**. The deduction from that theorem is elementary.

For positive bases, define

and

A Riemann-sum/lattice-point estimate gives

Let

This counts equal pairs of ordered representations. A non-permutation pair

has all six bases at most \(x^{1/k}\), so Salberger's theorem gives, for

\(k\ge26\),

Permutation pairs contribute \(6R_k(x)\), except that triples with repeated

coordinates have fewer than six distinct permutations. There are only

\(O_k(x^{2/k})\) such triples. Therefore

If \(F_k^+(x)=\#\{n\le x:r_k(n)>0\}\), Cauchy--Schwarz gives

and hence

On the other hand, the number of essentially unordered positive triples is

which is an upper bound for \(F_k^+(x)\). Thus

Allowing a zero base adds at most \(O_k(x^{2/k})\) essentially unordered

triples. Consequently the nonnegative version on the live page has the same

asymptotic:

This proves the stronger of the two requested bounds for every \(k\ge26\).

It also says almost every represented integer in this range has one

representation up to permutation.

Independent threshold arithmetic

For \(14 \[ \rho_d= \frac{12}{5}+\frac{27}{10\sqrt d}+\frac{9}{5d} -\frac{9}{20d\sqrt d}. \tag{2.7} \]

The checker recomputes with 50-digit decimal arithmetic that

It also checks all of Salberger's piecewise ranges and confirms that \(26\)

is the first integer where his published exponent drops below \(3\). For

\(d>34\), his exponent is \(131/45<3\).

For comparison, the checker independently evaluates the three exponents in

Browning--Heath-Brown and confirms that their first crossing is \(d=33\).

3. New extracted lower exponents for \(k=23,24,25\)

This section is **(b), rigorous modulo Salberger's published explicit

bound**.

Take

All \(B^3\) ordered triples with bases in \([1,B]\) have sum at most \(x\).

If \(E_k(B)\) is their equal-sum energy, Salberger gives

Cauchy--Schwarz therefore yields

For \(k=23,24,25\), \(\rho_k>3\), so

The exact formula (2.7) and independently recomputed decimals give the table

in the Outcome section. Salberger first improves Vaughan's

\((3-1/k)/k\) exponent at \(k=23\); the checker tests every \(7\le k\le25\)

and obtains exactly the improvement set \(\{23,24,25\}\).

At \(k=25\), the precise published wall is visible:

which loses \(0.0084\) in the \(B\)-exponent. At \(k=26\), the same machinery

crosses below \(B^3\), and the diagonal term takes over.

4. Exact computation at \(N=200\)

This entire section is (d), computational-only.

Let

For \(N=200\),

Every representation of an integer at most \(200^k\) has all bases at most

200. Thus sorting the sums from \(\mathcal T_{200}\), retaining those at most

\(200^k\), computes \(f_{k,3}(200^k)\) exactly.

The table counts \(0=0^k+0^k+0^k\). If “integers” is interpreted as positive

integers only, subtract exactly 1 from every entry.

| \(k\) | admissible unordered triples | \(f_{k,3}(200^k)\), including 0 |

|---:|---:|---:|

| 3 | 972,299 | 832,682 |

| 4 | 1,104,834 | 1,095,286 |

| 5 | 1,180,254 | 1,180,233 |

| 6 | 1,226,695 | 1,226,676 |

| 7 | 1,256,969 | 1,256,969 |

| 8 | 1,277,860 | 1,277,860 |

| 9 | 1,292,894 | 1,292,894 |

| 10 | 1,303,865 | 1,303,865 |

| 11 | 1,312,056 | 1,312,056 |

| 12 | 1,318,541 | 1,318,541 |

| 13 | 1,323,847 | 1,323,847 |

| 14 | 1,327,792 | 1,327,792 |

| 15 | 1,330,983 | 1,330,983 |

| 16 | 1,333,863 | 1,333,863 |

| 17 | 1,336,175 | 1,336,175 |

| 18 | 1,338,028 | 1,338,028 |

| 19 | 1,339,766 | 1,339,766 |

| 20 | 1,341,133 | 1,341,133 |

| 21 | 1,342,187 | 1,342,187 |

| 22 | 1,343,200 | 1,343,200 |

| 23 | 1,344,318 | 1,344,318 |

| 24 | 1,345,140 | 1,345,140 |

| 25 | 1,345,744 | 1,345,744 |

| 26 | 1,346,502 | 1,346,502 |

| 27 | 1,346,939 | 1,346,939 |

| 28 | 1,347,527 | 1,347,527 |

| 29 | 1,347,775 | 1,347,775 |

| 30 | 1,348,341 | 1,348,341 |

| 31 | 1,348,568 | 1,348,568 |

| 32 | 1,349,121 | 1,349,121 |

Full-box collision data

Here “excess” is

\(|\mathcal T_{200}|-\#\{a^k+b^k+c^k:(a,b,c)\in\mathcal T_{200}\}\).

| \(k\) | distinct full-box sums | excess | collision values | maximum unordered multiplicity |

|---:|---:|---:|---:|---:|

| 3 | 1,201,947 | 171,754 | 145,903 | 8 |

| 4 | 1,363,237 | 10,464 | 9,881 | 7 |

| 5 | 1,373,679 | 22 | 22 | 2 |

| 6 | 1,373,682 | 19 | 19 | 2 |

| every \(7\le k\le32\) | 1,373,701 | 0 | 0 | 1 |

Two primitive arithmetic checks found by the enumeration are

and

The verifier recomputes both sides and the gcds from scratch.

Why the large-\(k\) certificate is exact

For \(k=3,4,5,6\), the checker constructs and sorts the actual arbitrary

precision integer sums.

For \(7\le k\le32\), it constructs all sums modulo

It finds that all \(1,373,701\) residues are distinct for each \(k\). Equality

of two integer sums would force equality modulo \(M\). Therefore residue

injectivity is a deterministic proof of integer injectivity in this finite

box; no random-hash assumption or probable-prime assumption is involved.

The admissible-triple column is computed separately with exact integer powers

and binary search. The agreement of that count with the number of distinct

values for \(k\ge7\) then gives the exact \(f\)-values.

5. Reproduction

Classification: (d).

Standalone checker:

erdos325_wave8r_verify.py.

It uses only the Python standard library.

Run:

cd /home/exedev/MathDyad
python runs/erdos325_wave8r_verify.py

Observed result on this VM:

Literature-bound arithmetic: PASS
  d=23: rho=3.037170157679072, derived x-exponent=0.128818688796562
  d=24: rho=3.022307864403116, derived x-exponent=0.124070505649870
  d=25: rho=3.008400000000000, derived x-exponent=0.119664000000000
  d=26: rho=2.995350016380321, derived x-exponent=0.115384615384615
...
Exact enumeration and modular certificates: PASS
ALL CHECKS PASSED

The complete code is in the standalone file above. Its independent checks

include all hard-coded table values, both threshold crossings, the exact set

\(\{23,24,25\}\) where Salberger improves Vaughan, exact small-\(k\) collision

statistics, and a fresh modular injectivity certificate for every

\(7\le k\le32\).

6. Precise remaining wall

The energy reductions in this section are (a). Statements about the reach

of named published estimates are (b); the unpublished announcements are

(c); finite-search data and cost estimates are (d).

For a fixed \(3\le k\le25\), set

A sufficient missing lemma for the weaker live-page target is

Indeed Cauchy--Schwarz would then give

\(f_{k,3}(x)\gg_{k,\epsilon}x^{3/k-\epsilon}\). A

power-saving estimate for the non-permutation part,

would give the full asymptotic by the argument of Section 2.

The exact current obstructions found in this audit are:

• \(k=3\): unconditional smooth-Weyl-sum technology stops at

\(0.91709477<1\). Strong conjectures on Hasse--Weil \(L\)-functions can

reach the near-optimal energy, but they are conditional.

• \(4\le k\le22\): Vaughan's verified general exponent remains stronger

than the published determinant-method exponent found in this search.

• \(k=23,24,25\): Salberger is closest, but his energy exponent is still

respectively \(3.037170\ldots\), \(3.022307\ldots\), and \(3.0084\).

• Announcements are not proofs: the 2019 \(k\ge11\) workshop theorem and

2024 private \(k\ge16\) communication would be major improvements, but no

checkable proof was located. Using either would violate the requested

verifiability standard.

• Finite computation cannot supply uniformity: the \(N=200\) injectivity

certificate says nothing by itself about arbitrary \(B\). A naive extension

to \(N=1000\) has

\(\binom{1003}{3}=167,668,501\) residues, 1.34 GB before sort workspace for

one exponent; doing all 26 exponents would plausibly cost roughly 2--5

core-hours and several GB of RAM. Even completing that would not prove

(6.1). At \(N=10,000\), the raw 64-bit list alone is about 1.33 TB per

exponent.

Thus the mathematical task is not a larger finite search but a uniform

six-variable energy theorem below Salberger's published degree threshold.

PARTIAL: Published literature plus a checked deduction proves the full asymptotic for every k>=26, improves the lower exponents for k=23,24,25, and gives an exact N=200 table; the verifiable open range is k=3,...,25.