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Jaffe’s unrestricted successor-ratio conjecture is false

Abstract

The primitive Frobenius successor ratio exceeds five for an admissible pair of integers.

Definition 1.1 (Primitive nonnegative integer coefficients).

Formalization. D5/S3/ArithUnits/JaffeSuccessorRatioRefutation.Represented (✓ std3).

Citation. Grace Natalie Jaffe (2026). Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers. URL: https://arxiv.org/abs/2610.17267v1.

Commentary.

Represented(a,b,n) means that n = ax + by for nonnegative integers x and y with gcd(x,y)=1. The gcd is the natural gcd of their absolute values. Both coordinate axes are included. Passing between nonnegative integers and natural coefficients preserves the equality and coprimality.

Definition 1.2 (The full integer complement).

Formalization. D5/S3/ArithUnits/JaffeSuccessorRatioRefutation.missing (✓ std3).

Citation. Grace Natalie Jaffe (2026). Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers. URL: https://arxiv.org/abs/2610.17267v1.

Commentary.

The set missing(a,b) consists of all integers without a primitive nonnegative representation. Its domain is the whole integer line, including the negative integers.

Definition 1.3 (Greatest missing integer or infinity).

Formalization. D5/S3/ArithUnits/JaffeSuccessorRatioRefutation.sourceG (✓ std3).

Citation. Grace Natalie Jaffe (2026). Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers. URL: https://arxiv.org/abs/2610.17267v1.

Commentary.

sourceG(a,b) takes values in the extended real numbers. When the integer complement has a greatest element m, its value is the real embedding of m. When there is no greatest element, its value is positive infinity. For positive coprime A and B at least three, the counting argument below proves an actual greatest missing integer at least one.

Definition 1.4 (The next integer coprime to the first argument).

Formalization. D5/S3/ArithUnits/JaffeSuccessorRatioRefutation.successor (✓ std3).

Citation. Grace Natalie Jaffe (2026). Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers. URL: https://arxiv.org/abs/2610.17267v1.

Commentary.

successor(a,b) is the integer infimum of the integers c strictly greater than b and coprime to a. For a,b at least three this set is nonempty: a(b+1)+1 is greater than b and coprime to a. It is bounded below by b, so the integer infimum belongs to the set and is its least element.

Definition 1.5 (The whole unrestricted Conjecture 7.3).

Formalization. D5/S3/ArithUnits/JaffeSuccessorRatioRefutation.fullClaim (✓ std3).

Citation. Grace Natalie Jaffe (2026). Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers. URL: https://arxiv.org/abs/2610.17267v1.

Commentary.

For every pair of coprime integers a,b at least three, the claim bounds sourceG(a,successor(a,b))/sourceG(a,b) by five. No relation a<b is assumed. The coefficients, complement and successor have exactly the meanings given above. The same assertion is Conjecture 8.3 in the earlier thesis.

Theorem 1.6 (The full conjecture has a counterexample).

Proof. Machine-checked in Lean as D5/S3/ArithUnits/JaffeSuccessorRatioRefutation.result (✓ std3). ∎

Resolves. Problems/jaffe-primitive-frobenius-successor-ratio (refuted) by D5/S3/ArithUnits/JaffeSuccessorRatioRefutation.result.

Source. Repository-derived.

Acknowledgement. Grace Natalie Jaffe (2026). Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers. URL: https://arxiv.org/abs/2610.17267v1.

Commentary.

Take a=87841397512641630 and b=7. The integer a is coprime to 7 and 49, and each integer from 8 through 48 is noncoprime to a. Hence successor(a,7)=49. Let H(n) count all nonnegative solutions of Ax+By=n, and let P(n) count the primitive ones. Residues modulo B place H(n) between floor(n/(AB)) and floor(n/(AB))+1. Partitioning by gcd(x,y) and applying Mobius inversion gives AB P(n) >= phi(n)-AB tau(n), where tau(n) is the positive-divisor count. Factoring phi and tau over prime powers gives n tau(n)^2 <= 16^7 phi(n)^2. For primes at least seven the corresponding factor has constant one; a factor sixteen suffices for each smaller prime, of which there are at most seven. Therefore every n>16^7(AB)^2 has a primitive representation. Since 1 is missing when A,B>=3, the entire integer complement has a greatest element at least one. Apply this to (a,7) and (a,49), obtaining actual finite positive maxima m and M. A primitive representation of 7m for (a,49) would force its first coefficient to be divisible by 7. Dividing that coefficient preserves coprimality and would represent m for (a,7), a contradiction. Thus M>=7m and M/m>=7>5. The quotient uses these actual maxima; their numerical values are unnecessary. Finiteness and scaling are prior ingredients of the source’s Theorems 4.3 and 7.1. The added condition a<b would define a different assertion, whose truth is not established by this counterexample.

References