Erdős 699: asymmetric prime-pair bounds

P699 research · Erdős problem 699 · Numerical BFT bounds

Bennett, Filaseta and Trifonov take a symmetric maximum near the end of their Padé argument. Retaining the two prime-power logarithms separately gives different exponents for the two cofactors. For example, their Proposition 5.1 parameters give the entirely numerical assertion

Here $u,v$ are positive integers and $k,l$ are nonnegative integers. No coprimality assumption is required. The two inequalities in the conclusion form a disjunction, not a simultaneous assertion.

The source is BFT, Section 7, in the author preprint associated with On the factorization of consecutive integers. The refinement and threshold calculation below are deductions in this repository.

1. The asymmetric exponents

Use any BFT Theorem 2.4 parameter choice. Write

and suppose its coefficient-gcd bound

has been proved. Assume $\Omega_3>1$ and $\Omega_4>1$, with the notation of that theorem. Set

The limiting cofactor exponents are

Thus the larger seed prime power retains the symmetric BFT exponent; the smaller seed prime power has a strictly larger cofactor exponent. For every fixed bounded gap and every positive $\epsilon$, the conclusion

holds beyond an effectively computable threshold. The next section gives a fully numerical formula when $L_1,m_0$ are explicit.

The same effective conclusion holds with the limiting coefficient bound $L(s)$ from BFT Theorem 2.2. Its Section 7 proof supplies an effective $m_0$ after replacing $L(s)$ by $L(s)^{1-\epsilon'}$. Choose $\epsilon'>0$ sufficiently small to preserve any strictly smaller target exponents in (3). This does not assert a numerical threshold until that coefficient-gcd estimate is instantiated.

2. A direct numerical threshold

Fix a gap bound $D\ge1$. Let $\kappa_1,\kappa_2$ be the explicit positive constants in BFT Theorem 2.4, for this $D$. Choose target exponents $0<t_p<\alpha_p$ and $0<t_q<\alpha_q$, and define

Put

For positive integers $u,v$ and nonnegative integers $k,l$, suppose

Then

Proof. Suppose both conclusions fail. BFT's auxiliary integer is

The assumed cofactor upper bounds imply $p^k\ge x^{1-t_p}$ and $q^l\ge x^{1-t_q}$. Consequently

Divide the original two prime powers into their blocks of length $cm$:

The integer linear combination in BFT (7.4), together with $m>M$, gives its estimate (7.6):

Multiplying by $Q^{cm}$ and using (2) yields

This step only uses the absolute difference bound in the integer linear combination. It does not require that $p^ku$ be the larger input, and it does not use BFT (7.7) or a coprimality hypothesis on the cofactors. The nonvanishing choice between the two Padé approximants is exactly the determinant argument preceding (7.4).

Since $x>D$, we also have $q^lv\le x+D<2x$. Combining this with (7) and (8) gives

contradicting the last term of (5). This proves (6).

Formula (5) uses the cofactor upper bounds directly. It does not incur the additional estimates used to pass from BFT's auxiliary parameter to one symmetric maximum of the two cofactors.

3. The pair $5,7$

Use the BFT seed and Proposition 5.1 entry

Directed interval arithmetic gives

For $D=100$, formula (5) yields these numerical statements:

For the first row, the controlling quantity is $cL(M+1)<2101.583$; the other nontrivial term is less than $1115.615$. The final certificate verifies $H<913\log10$ directly.

These inequalities can be combined with all the original symmetric prime-pair constraints: each supplies an additional disjunction whose two endpoints now have different weights. Replacing it by the smaller of the two weights would discard the new information.

4. A smaller numerical last row at index eleven

Combine (1) with the six original BFT Theorem 2.1 constraints on $2,3,5,7$. The index-eleven weighted argument then has exact minimum gain

where the minimum ranges over all endpoint choices for the seven disjunctions. It is attained by $(v_2,v_3,v_5,v_7)=(.285,0,.29,.038)$ and $(.285,.038,.29,0)$. The checker verifies all $2^7$ endpoint choices with exact rational arithmetic.

Retaining the coefficients in the weighted argument improves the numerical comparison. For weights $m_t=14-t$, the product of the largest possible off-maximum small-prime contributions is

The exact weighted estimate, for $n\ge945000$, is

The last comparison is checked with integers. The checker also considers every possible collision of maximal positions using the weighted row capacities; the minimum gain remains $13/125$. For $n\ge10^{914}$, all $n-r_p$ exceed $10^{913}$, so the new asymmetric constraint and the original BFT numerical constraints apply. This contradicts the algebraic bound. Consequently

This replaces the exponent $3\,639\,160\,902$ obtained from the stronger symmetric cofactor table. It is a complete numerical stopping bound, with no unevaluated source constants.

The checker and recorded intervals use the existing exact polynomial-integral evaluator for $\kappa_1,\kappa_2$, and 80-digit directed intervals for all logarithms. No rows are enumerated.

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_bft_asymmetric.py

5. Moderate numerical bounds at indices 13, 16 and 21

The same direct threshold formula applied to the seed $8-7=1$ with the published parameters $c/d=3/2$, $L_1=1.5395$, $m_0=138$ gives

Its limiting endpoint exponents exceed $.25923834$ and $.30680637$. The exact interval certificate is included in the same source checker. Combining the asymmetric alternatives with the original explicit BFT table and the weighted polynomials gives:

Each final cap includes all source thresholds and numerator shifts. The two symmetric pairs in the last row have the previously certified numerical thresholds $10^{976}$ and $10^{957}$.

The exact weighted checker and its output verify all minimal disjunction vectors, all collisions of maximal positions, and the integer comparisons establishing these bounds. In particular, index thirteen has a moderate numerical bound without needing the interpolation coefficient estimate.