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Guided by the Primes--An Exploration of Very Triangular Numbers

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Guided by the Primes--An Exploration of Very Triangular Numbers GUIDED BY THE PRIMES - AN EXPLORATION OF VERY TRIANGULAR NUMBERS AUDREY BAUMHECKEL AND TAMAS´ FORGACS´ “Adding up the total of a love that’s true / Multiply life by the power of two.” - Indigo Girls Abstract. We present a string of results concerning very triangular numbers, paralleling fundamental theorems established for prime numbers throughout the history of their study, from Euclid (∼300 BCE) to Green-Tao (2008) and beyond. 1. Introduction Our investigation was prompted by one of an anthology of essays that considers a handful of engaging topics in mathematics [6]. The essay considers very triangular numbers, which are triangular numbers whose binary representation contains a n(n+1) triangular number of 1s. While the formula tn = 2 for the nth triangular number is known, to the best of our knowledge no explicit formula is known for the nth very triangular number. One can take the lack of such as a first parallel between the theory of very triangular numbers and the theory of prime numbers. To our delight, it turns out that there are several other results regarding prime numbers which have analogues in the theory of very triangular numbers. The ones considered in this paper (in the order they are presented) are: the infinitude of primes, arithmetic progressions and related results including the twin prime conjecture, and the existence of strings of consecutive numbers that either must, or do not contain primes. Following the sections containing our results, we conclude the paper with some open problems and potential further avenues of investigation. In the remainder of the paper we shall use the the following notational conventions: arXiv:2105.10354v1 [math.HO] 19 May 2021 T denotes the set of triangular numbers; VT denotes the set of very triangular 1 numbers, and |(n)2| denotes the number of 1s in the binary representation of the positive integer n. 2. The infinitude of very triangular numbers One of the oldest results regarding the prime numbers concerns their infinitude. As is well known, Euclid’s theorem (Proposition 20 in Book IX of the Elements) states that no finite list of prime numbers contains all primes. Many of the sub- sequent proofs are similar to Euclid’s in the sense that they establish the infini- tude of primes in a non-formulaic way. There are however proofs (most recently in this Monthly by Elsholtz, see [2]) that provide prime representing functions f : N →{prime numbers}. In priming the reader for further inquiry, Tanton [6, p.217, Theorem] shows that 1 2 AUDREYBAUMHECKELANDTAMAS´ FORGACS´ n 1 there are infinitely many very triangular numbers, by noting that |(t2 +3)2| =6 ∈ T for all n ∈ N. One can readily extend his argument as follows. Theorem 2.1 (Infinitude of VT -II). Suppose that 2(ℓ + 1) is an even triangular number, and let n> 2ℓ−1. Then t2n+2ℓ−1 ∈ VT . In particular, there are infinitely many very triangular numbers. Proof. Note that (2n + ℓ−1 2i)(2n +2ℓ) ℓ−1 i=0 = (2n + 2i)(2n−1 +2ℓ−1) 2 i=0 P X ℓ−1 ℓ−1 = 22n−1 +2n+ℓ−1 + 2n−1+i + 2ℓ−1+i. i=0 i=0 X X The restriction on n ensures that each power of 2 appearing in the last expression is distinct (2n − 1 > n + ℓ − 1 > n + ℓ − 2 > ··· >n − 1 > 2ℓ − 2 > ··· > ℓ − 1). Since there are exactly 2(ℓ + 1)-many distinct powers, we conclude that 1 n ℓ (t2 +2 −1)2 = 2(ℓ + 1) ∈ T, and consequently, t2n+2ℓ−1 ∈ VT as claimed. Theorem 2.1 shows that given any even triangular number k, there are infinitely many very triangular numbers with k many 1s in their binary representation. Our next result shows that given any triangular number k, there are very triangular numbers with k many 1s in their binary representation. Theorem 2.2 (Infinitude of VT -III). Let k ∈ T be a triangular number. Then k ℓ k k ℓ 1 k t2 −2 ∈ VT for all 0 ≤ ℓ ≤ 2 . Moreover, |(t2 −2 )2| = k for all 0 ≤ ℓ ≤ 2 . Proof. Let k ∈ T be given. The cases i =0, 1 are treated in Theorem 3.4. Suppose k now that 2 ≤ ℓ ≤ 2 . We compute k ℓ k ℓ (2 −2 )(2 − 2 + 1) 1 k+ℓ+1 k−ℓ−1 ℓ k−ℓ 2ℓ t k ℓ = = 2 (2 − 1)+2 (2 − 1)+2 2 −2 2 2 k−ℓ−2 k−ℓ−1 = 2k+ℓ+j + 2i+ℓ−1 +22ℓ−1 j=0 i=0 X X k−ℓ−2 ℓ−1 k−ℓ−1 = 2k+ℓ+j + 2i+ℓ−1 + 2i+ℓ−1 +22ℓ−1 j=0 i=0 X X Xi=ℓ k−ℓ−2 ℓ−1 k+ℓ+j i+ℓ−1 k−1 = 2 + 2 +| 2{z . } j=0 i=0 X X k Given the restriction 2 ≤ ℓ ≤ 2 we see that all powers of 2 in the last expression are distinct. Since there are (k − ℓ − 1)+ ℓ +1= k many of these, we conclude that 1 |(t2k−2ℓ )2| = k, as claimed. One may wonder whether given an odd triangular number, are there very tri- angular numbers with that many 1s in their binary representation, and if so, how 1 many? Setting ℓ = 0 in Theorem 2.2 shows that |(t2k−1)2| = (2k−2)−(k−1)+1 = k, hence the answer to the first question in in the positive. The answer to the second GUIDED BY THE PRIMES - AN EXPLORATION OF VERY TRIANGULAR NUMBERS 3 question – as presented in the following two propositions – is perhaps somewhat surprising. While there are infinitely many very triangular numbers vt ∈ VT with 1 |(vt)2| = 15, 21, 45, 55,...(all odd triangular numbers greater than 3), we are rea- sonably certain that there are only four very triangular numbers with three 1s in their binary representations. Proposition 2.3. Let ℓ > 1 so that 2ℓ +1 is a triangular number. There are 1 infinitely many very triangular numbers vt ∈ VT with |(vt)2| =2ℓ +1. Proof. Given ℓ as in the statement, consider the triangular numbers t22ℓ−2ℓ+1. We compute 1 2ℓ ℓ 2ℓ ℓ t 2ℓ ℓ = 2 − 2 +1 2 − 2 +2 2 −2 +1 2 = 24ℓ−1 − 23ℓ +22ℓ +1 − 2ℓ − 2ℓ−1 +1 ℓ−2 ℓ = 23ℓ+i + 2ℓ+i − 2ℓ−1 +1 i=0 i=0 X X ℓ−2 ℓ = 23ℓ+i + 2ℓ+i +2ℓ−1 +1. i=0 i=1 X X Given the restriction on ℓ, the the powers of 2 in the above summands are all distinct, and there are (ℓ − 1)+ ℓ +1+1=2ℓ + 1 many of them. We complete this section with a discussion of triangular numbers with three 1s in their binary representations. N 1 1 Conjecture 2.4. If n ∈ is such that |(n)2|≥ 6, then |(tn)2|≥ 4. 1 1 Proposition 2.5. The only very triangular numbers tn with |(n)2|≤ 5 and |(tn)2| = 3 are 21, 28, 276 and 1540. 1 1 1 Proof. The case of |(n)2| = 1 trivially yields |(tn)2| = 2 6= 3. The cases |(n)2| = 2, 3, 4, 5 are all handled by case analysis. Lest we fall victim of paralysis by (sub)- 1 case analysis, we only present here the case |(n)2| = 3, as this case is ‘big enough’ to be representative of the others, and to indicate the general method of reasoning; yet small enough not to cause the reader any undue discomfort. Suppose thus 1 σ(1) σ(2) σ(3) that |(n)2| = 3, and write n = 2 +2 +2 for some increasing function σ : {1, 2, 3} ֒→ N∪{0}. We will need to distinguish between the cases when σ(1) = 0 and when σ(1) > 0. Consider first the case σ(1) ≥ 1, and the corresponding diagram (see Figure 1) representing the summands in n(n + 1), with solid arrows in the diagram pointing towards known larger quantities, and dashed ones between quantities we cannot compare without further assumptions. If σ(2) > 2σ(1), then the sum looks like 2 + 22σ(1) +2σ(2)+ terms whose powers 1 are all greater than σ(2). As such, we conclude that |(n(n + 1))2|≥ 4. If σ(2) = 2σ(1), then (2.1) n(n +1)=2σ(1) +22σ(1)+1 +23σ(1)+1 +24σ(1) + terms with powers involving σ(3). We now consider the subcases σ(1) = 1 and σ(1) > 1. In the former, we obtain n(n +1)=2+23 +25 + terms with powers involving σ(3). 4 AUDREYBAUMHECKELANDTAMAS´ FORGACS´ 2σ(1) + 2σ(2) + 2σ(3) 22σ(1) + 2σ(1)+σ(2) + 2σ(1)+σ(3) 2σ(1)+σ(2) + 22σ(2) + 2σ(2)+σ(3) 2σ(1)+σ(3) + 2σ(2)+σ(3) + 22σ(3) Figure 1. 1 The case |(n)2| = 3 and σ(1) ≥ 1. The smallest of the remaining powers is σ(3) ≥ 3. If σ(3) = 4 or σ(3) ≥ 6, we 1 1 trivially obtain that |(n(n + 1))2| ≥ 4. If σ(3) = 3, we get |(n(n + 1))2| = 4, 1 while if σ(3) = 5, we obtain |(n(n + 1))2| = 6. If on the other hand σ(1) > 1, then the four displayed powers of 2 in (2.1) are all distinct. We must therefore have σ(3) = 2σ(1)+ 1, or σ(3) = 3σ(1) + 1, as all other cases immediately yield 1 |(n(n + 1))2|≥ 4. Suppose first that σ(3) = 2σ(1)+1.
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