The world of reality has its limits; The world of imagination is boundless. — Jean Jacques Rousseau
Human imagination has its limits; The world of reality is boundless. — Kyriakos Kefalas (2025)
On Arithmetic Progressions Divisible by a Twin Prime Product
The Problem
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Find ‘clean’ arithmetic progressions (AP) of integers such that all its terms are divisible by a twin prime product. Clean here means that no other integer lying between the terms of the AP, is divisible by a twin prime product.
Solutions
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Such AP with 6, terms or more are rare. The rarity increases significantly with the number of terms. We present the smallest examples for a number of terms.
5 Terms AP
The smallest 5–term clean AP whose terms are divisible by a twin prime product are
There are no other such 5–term clean AP below, \(x\leq 1000\). Since the linear coefficient is divisible by the twin prime product, \(3\times 5=15\), it is not unexpected that the same twin pair divisor divides all terms in this case. In all 3, cases, the middle term is divisible by the only triplet of adjacent primes that exists, \((3,\,5,\,7)\). This is because the twin prime product, \(5\text{[Cross]}7=35\), must divide one of the terms of the AP, otherwise it would contradict the clean property of the AP. Since a 2nd integer divisible by 35, cannot belong to the AP, the only suitable term that can be divided by 35, is the middle term.
Lemma
There exists no clean AP with linear coefficient 15, which has more than 5, terms.
Proof: The interval spanned by such an AP would be at least 90 (for 6, terms), hence two adjacent multiples of 35, have to be both, terms of the AP, otherwise the AP is not clean. But this is impossible for linear coefficient 15. ■■
Lemma
There exists no clean AP with linear coefficient larger than 15.
Proof: Since the smallest twin prime product is 15, any linear coefficient larger than 15, would contradict the clean property of the AP. ■■
5 Terms AP, linear coefficient less than 15
In this case the smallest 5–term AP satisfying all the conditions of the problem is
There are no other such 6–term AP below, \(x\leq {10}^{8}\).
Open task
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The task is to find clean AP that satisfy the conditions of the title problem with more terms.
What was “Unexpected Mathematics” in this problem
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It was not expected that finding clean AP that satisfy the simple conditions of the title problem, would be so hard computationally. We really had to stretch our resources and still have found no large AP which satisfies the conditions of the problem.
Twin prime products
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For reference we present all twin prime products below, \(p\,(p+2)\leq 1200000\).
Unexpected Mathematics, do not imply that the displayed mathematics are false or unprovable. It only denotes that they are counter–intuitive. They are also usually not a straightforward consequence of a well known and established fact. They are nevertheless proved mathematical facts in the form they are displayed, whose underlying reason or more general proof may be unknown.
“To boldly go where no one has gone before.” — Gene Roddenberry
