Goldbach Conjecture: Violation Probability and Generalization to Prime-like Distributions

Ameneh Farhadian

REVIEW

Goldbach Conjecture: Violation Probability and Generalization to Prime-like Distributions

Ameneh Farhadian

Number Theory · Submitted 19 Apr 2025 · 8 references

Pith's verdict: The generalized Goldbach claim is refuted by a residue-class obstruction, the probability derivation is heuristic with an undefined model and a sign error, but the clean exposition and the shifted-prime observation have some value.

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arXiv ID2504.14353 (version 1)

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CitationAmeneh Farhadian. "Goldbach Conjecture: Violation Probability and Generalization to Prime-like Distributions." arXiv:2504.14353 (2025). https://pith.science/paper/NTVRLMF4

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Claim paper

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Short version: the main theorem is not just unproven; it is false for the class of sets it claims to cover. The bound e^{-N^α} rests on a heuristic that treats deterministic sets as random, and the algebra in (19) has the sign backwards. A residue-class obstruction kills the generalized conjecture entirely.

What the paper genuinely does: it restates Clarke and Shannon's combinatorial heuristic in a cleaner way, introduces the notion of a prime-like distribution, and checks the heuristic numerically for shifted primes. The shifted-prime theorem (Theorem 1) is true but trivial: if Goldbach holds for primes, shifting both summands by the same constant c gives 2n+2c. The exponential bound with α ≈ 1 - 2lnln N/ln N is new as a heuristic.

The problems are serious.

First, Definition 1 is far too permissive. The counting error bound |π_Q(n)-π(n)|

Second, even for the primes, the derivation is heuristic, not a proof. Lemma 1 treats the deterministic sets A_n and B_n as independent uniform subsets of {0,...,n-1}; no probability space is defined that would justify the hypergeometric count (9).

Third, equation (18) has the wrong sign. From n/ln^2 n > n^α + (α-1)ln n + ln α one obtains e^{-n/ln^2 n} < (1/α) e^{-n^α} n^{1-α}, not α e^{-n^α} n^{α-1}. The claimed integral evaluation does not follow.

The computational check for random ±1 shifts is a reasonable sanity check, but the code and the exact subsets are not provided, so it is not reproducible.

Who is this for? A reader who wants a cleaned-up exposition of the Clarke-Shannon heuristic could skim it. As a mathematical proof, it fails. My recommendation: desk reject. The counterexample is elementary, and the remaining derivation is not rigorous.

Referee report

Major objections (4)

[Section 3, Lemma 1, Eq. (9)] The sets A_n and B_n defined in Eqs. (7)-(8) are deterministic functions of Q, but Eq. (9) computes the probability that a uniformly random k_2-subset of {0,...,n-1} is disjoint from a fixed k_1-subset. No probability measure on Q, on A_n, or on B_n is ever defined, and the independence used to pass from Eq. (9) to Eq. (10) is assumed without justification. This missing probability space is load-bearing: every subsequent bound, including Lemma 1, Theorem 2, and Theorem 3, inherits the defect.
[Section 2, Definition 1 and Conjecture 1] Conjecture 1 is false as stated. Definition 1 allows prime-like sets with a fixed additive obstruction: one can construct a set Q by mapping the primes injectively via a bounded displacement into a fixed residue class modulo 3. Such a Q satisfies |\pi_Q(n)-\pi(n)| \le C for a constant C, so it is prime-like, but every element of Q is congruent to 1 mod 3. Hence every sum of two elements is congruent to 2 mod 3, and no even integer 2n \equiv 1 mod 3 (equivalently n \equiv 2 mod 3) can be represented, for infinitely many n. For these n the failure probability is 1, contradicting Lemma 1's bound exp(-n/\ln^2 n).
[Section 3, Eqs. (18)-(19)] Even if the unproved inequality (18) were granted, the deduction of Eq. (19) is algebraically incorrect. From n/\ln^2 n > n^\alpha + (\alpha-1)\ln n + \ln \alpha it follows that e^{-n/\ln^2 n} < (1/\alpha) e^{-n^\alpha} n^{1-\alpha}, not \alpha e^{-n^\alpha} n^{\alpha-1}. The two expressions differ by the factor (1/\alpha^2)n^{2-2\alpha}, which tends to infinity when \alpha<1. Consequently Eq. (20) and the claimed e^{-N^\alpha} bound do not follow from the displayed inequalities. Inequality (18) itself is also asserted without proof; the 'growth rate' statement is not a derivation.
[Section 3, Theorems 2 and 3] The proof sums the quantities P(n) over n \ge N as if a union bound applied to failure events in a single probability space. Even if each P(n) were a valid probability, the sum is an upper bound for the probability of the union only if all events live in one common probability space, and no such space is constructed. The phrase 'assuming the Goldbach conjecture holds for all even numbers less than 2N' is a deterministic verification statement, not a conditioning event in a probabilistic model, so it cannot by itself turn the heuristic calculation into a probability bound.

Minor objections (4)

[Abstract and Section 3, Theorem 2] The abstract and Theorem 3 state a bound e^{-N^\alpha}, while Theorem 2 states e^{-\sqrt{N}}; the two statements are not reconciled, and the proof of Theorem 2 is then superseded by Theorem 3 without comment.
[Section 2, Eq. (4)] For an integer shift t, the bound should be |\pi(n-t)-\pi(n)| < |t|+1, not t+1, since t may be negative.
[Section 2, random \pm1 shifts] The phrase 'By randomly addition of +1 and -1 to prime numbers' is imprecise: Definition 1 applies to a fixed set, not to a random construction, and the claimed discrepancy bound <2 should be stated for individual realizations with a precise interval argument.
[Throughout] There are numerous typographical errors, including 'Altough', 'previus', 'mare', 'gratear than N0', and 'Corrollary'; these should be corrected in any revision.

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The paper

The paper claims

If the Goldbach conjecture holds through $2N$, the probability of any later counterexample is below $e^{-N^{\alpha}}$.

The reading

According to the paper, once Goldbach's conjecture has been verified for every even number below $2N$, the probability that any larger even number fails to be a sum of two primes is at most $e^{-N^{\alpha}}$, where $\alpha = 1 - 2\ln\ln N/\ln N$. For the range already checked, $N = 4\times 10^{18}$, this places the failure probability below $e^{-10^{15}}$, which the author describes as making counterexamples practically nonexistent. The calculation never inspects whether individual numbers are prime; it only uses the fact that primes near $n$ occur with density about $1/\ln n$. The paper therefore extends the same conclusion to any subset of integers whose counting function stays within a bounded distance of $\pi(n)$, and reports computational verification of that generalized Goldbach property for randomly shifted prime sets up to $2\times 10^{8}$. The contribution is a probabilistic, distribution-based reformulation of Goldbach rather than a proof that every even number has a representation.

What carries the argument

The mechanism is the pair of distance sets $A_n$ and $B_n$ defined inside $\{0,\ldots,n-1\}$. Their intersection is exactly the set of equal distances that place $n$ midway between two elements of $Q$, so Goldbach's condition for $2n$ is $A_n\cap B_n\neq\varnothing$. The paper estimates the sizes of both sets as $n/\ln n$ from the prime distribution, computes the hypergeometric probability that two subsets of those sizes are disjoint, bounds that probability by $e^{-n/\ln^2 n}$, and converts the sum of these tail probabilities from $n=N$ to infinity into $e^{-N^{\alpha}}$ by an integral estimate. The same machinery carries over to any set with a prime-like counting function, which is why the paper can generalize the conjecture without testing primality.

Extended reading notes

Core claim

The paper's central assertion is that failure of the Goldbach conjecture is governed by the density of primes, not by their specific identities. For an even number $2n$, define $A_n$ as the set of distances $n-q$ for elements $q\in Q$ at or below $n$, and $B_n$ as the distances $q-n$ for elements $q\in Q$ between $n$ and $2n$. A representation $2n=q_1+q_2$ exists exactly when $A_n\cap B_n$ is nonempty. Taking both sets to be random subsets of $\{0,\ldots,n-1\}$ of size roughly $n/\ln n$, the paper bounds the probability of disjointness by $\exp(-n/\ln^2 n)$ and, summing over all $n\ge N$, obtains the tail bound $e^{-N^{\alpha}}$. Because the calculation depends only on the counting estimate $\pi_Q(n)\approx n/\ln n$, the paper states the result for every prime-like set $Q$, proves that shifted prime sets $P_c=\{p+c\}$ inherit Goldbach from the primes, and proposes the generalized conjecture that every such set eventually represents all sufficiently large even integers.

Load-bearing premise

The load-bearing premise is that the two fixed sets of distances $A_n$ and $B_n$ can be treated as independent random subsets of $\{0,\ldots,n-1\}$, so that the hypergeometric formula in equation (9) is a genuine probability that Goldbach fails for $2n$; in the paper these sets are deterministic functions of $Q$, and no probability measure over counterexamples is defined.

Editorial extensions

If this is right

Substituting the verified range $N=4\times10^{18}$ yields the paper's numerical conclusion: the probability of a remaining counterexample is less than $e^{-10^{15}}$.
Any future extension of the verification range automatically strengthens the bound, since the exponent $N^{\alpha}$ increases with $N$.
For every prime-like set $Q$ satisfying the definition, the same conditional tail bound applies, giving the generalized Goldbach conjecture the same probabilistic support primes receive.
Shifted prime sets $P_c=\{p+c\}$ are provably covered by the ordinary Goldbach conjecture if the latter holds, and random $\pm1$ shifts of primes were verified for all even integers up to $2\times10^{8}$, supporting the generalized conjecture in practice.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A formally complete version of the argument would need to define the probability space; one natural reading is a random model in which each integer near $n$ is included in $Q$ independently with probability about $1/\ln n$, under which the expected intersection size $|A_n\cap B_n|$ grows like $n/\ln^2 n$ and disjointness becomes rare. The paper itself does not define such a measure.
The distance-set mechanism is not specific to sums of two primes: the same equal-distances-around-$n$ idea could be applied to other additive configurations whenever a density law is known, though the paper does not pursue this.
A natural stress test beyond the paper's computations is to preserve the density condition while altering primes in other ways, for example by randomly deleting a sparse set of primes or applying slowly varying shifts, and check whether the generalized Goldbach property persists numerically.

Original abstract

Due to the distribution of primes among integers, we establish an upper bound for the probability $\mathbb{P}_n$ that the Goldbach conjecture fails. Assuming the conjecture holds true for all even number less than $2N$, we prove this probability is less than $e^{-N^\alpha}$, where $ \alpha = 1 - \frac{2\ln\ln N}{\ln N}$. For large $N$, this probability becomes vanishingly small, effectively precluding the existence of counterexamples in practice. If $N =4 \times 10^{18}$, the probability of a counterexample is less than $e^{-10^{15}}$. Our approach fundamentally depends on the distributional properties of primes rather than their primality per se. This perspective enables a natural generalization of the conjecture to non-prime subsets of integers that exhibit similar distributional characteristics. As a concrete example, we construct new subsets by applying random $\pm 1$ shifts to primes, which preserve the essential prime-like distributional properties. Computational verification confirms that this generalized Goldbach conjecture holds for all even integers up to $2 \times 10^{8}$ within these modified subsets.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,360 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.27 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.13 · states quantitative result 0.98 · compares with prior results 0.08 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Due to the distribution of primes among integers, we establish an upper bound for the probability $\mathbb{P}_n$ that the Goldbach conjecture fails. Assuming the conjecture holds true for all even number less than $2N$, we prove this probability is less than $e^{-N^\alpha}$, where $ \alpha = 1 - \frac{2\ln\ln N}{\ln N}$. For large $N$, this probability becomes vanishingly small, effectively precluding the existence of counterexamples in practice. If $N =4 \times 10^{18}$, the probability of a counterexample is less than $e^{-10^{15}}$. Our approach fundamentally depends on the distributional properties of primes rather than their primality per se. This perspective enables a natural generalization of the conjecture to non-prime subsets of integers that exhibit similar distributional characteristics. As a concrete example, we construct new subsets by applying random $\pm 1$ shifts to primes, which preserve the essential prime-like distributional properties. Computational verification confirms that this generalized Goldbach conjecture holds for all even integers up to $2 \times 10^{8}$ within these modified subsets.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Due to the distribution of primes among integers, we establish an upper bound for the probability $\mathbb{P}_n$ that the Goldbach conjecture fails.	contribution 0.71	0.88	0.19	asserted 1.00
s1	Assuming the conjecture holds true for all even number less than $2N$, we prove this probability is less than $e^{-N^\alpha}$, where $ \alpha = 1 - \frac{2\ln\ln N}{\ln N}$.	result 0.97	0.90	0.32	asserted 1.00
s2	For large $N$, this probability becomes vanishingly small, effectively precluding the existence of counterexamples in practice.	interpretation 0.98	0.75	0.33	asserted 0.95
s3	If $N =4 \times 10^{18}$, the probability of a counterexample is less than $e^{-10^{15}}$.	result 0.99	0.82	0.14	asserted 1.00
s4	Our approach fundamentally depends on the distributional properties of primes rather than their primality per se.	interpretation 0.43	0.68	0.08	asserted 1.00
s5	This perspective enables a natural generalization of the conjecture to non-prime subsets of integers that exhibit similar distributional characteristics.	contribution 0.64	0.76	0.18	asserted 0.86
s6	As a concrete example, we construct new subsets by applying random $\pm 1$ shifts to primes, which preserve the essential prime-like distributional properties.	method 0.76	0.91	0.04	asserted 1.00
s7	Computational verification confirms that this generalized Goldbach conjecture holds for all even integers up to $2 \times 10^{8}$ within these modified subsets.	result 1.00	0.89	0.10	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (8)

[1]

JH Clarke and AG Shannon. 67.2 a combinatorial approach to goldb ach conjecture. The Mathematical Gazette , 67(439):44–46, 1983

View on Pith 1983
[2]

Theodor Estermann. On goldbach’s problem: Proof that almost a ll even positive integers are sums of two primes. Proceedings of the London Math- ematical Society, 2(1):307–314, 1938

View on Pith 1938
[3]

Ameneh Farhadian and Hamid Reza Fanai. A simple explanation for th e goldbach conjecture. arXiv preprint arXiv:2211.02865 , 2022

View on Pith Pith review arXiv 2022
[4]

Godfrey H Hardy and John E Littlewood. Some problems of ‘partitio numerorum’; iii: On the expression of a number as a sum of primes. Acta mathematica, 44(1):1–70, 1923

View on Pith 1923
[5]

Kenneth Ireland, Michael Ira Rosen, and Michael Rosen. A classical introduction to modern number theory , volume 84. Springer Science & Business Media, 1990

View on Pith 1990
[6]

H Montgomery and R Vaughan. The exceptional set of goldbach’s prob- lem. Acta arithmetica, 27:353–370, 1975

View on Pith 1975
[7]

Tom´ as Oliveira e Silva, Siegfried Herzog, and Silvio Pardi. Empirical veriﬁcation of the even goldbach conjecture and computation of p rime gaps up to 4 ×1018. Mathematics of Computation , 83(288):2033–2060, 2014

View on Pith 2014
[8]

Yuan Wang. The Goldbach Conjecture , volume 4. World scientiﬁc, 2002. 11

View on Pith 2002

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What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Discussion

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[1] [1]

JH Clarke and AG Shannon. 67.2 a combinatorial approach to goldb ach conjecture. The Mathematical Gazette , 67(439):44–46, 1983

View on Pith 1983

[2] [2]

Theodor Estermann. On goldbach’s problem: Proof that almost a ll even positive integers are sums of two primes. Proceedings of the London Math- ematical Society, 2(1):307–314, 1938

View on Pith 1938

[3] [3]

Ameneh Farhadian and Hamid Reza Fanai. A simple explanation for th e goldbach conjecture. arXiv preprint arXiv:2211.02865 , 2022

View on Pith Pith review arXiv 2022

[4] [4]

Godfrey H Hardy and John E Littlewood. Some problems of ‘partitio numerorum’; iii: On the expression of a number as a sum of primes. Acta mathematica, 44(1):1–70, 1923

View on Pith 1923

[5] [5]

Kenneth Ireland, Michael Ira Rosen, and Michael Rosen. A classical introduction to modern number theory , volume 84. Springer Science & Business Media, 1990

View on Pith 1990

[6] [6]

H Montgomery and R Vaughan. The exceptional set of goldbach’s prob- lem. Acta arithmetica, 27:353–370, 1975

View on Pith 1975

[7] [7]

Tom´ as Oliveira e Silva, Siegfried Herzog, and Silvio Pardi. Empirical veriﬁcation of the even goldbach conjecture and computation of p rime gaps up to 4 ×1018. Mathematics of Computation , 83(288):2033–2060, 2014

View on Pith 2014

[8] [8]

Yuan Wang. The Goldbach Conjecture , volume 4. World scientiﬁc, 2002. 11

View on Pith 2002