On the Largest Integer that is not a Sum of Distinct Positive nth Powers

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1 On the Largest Integer that is not a Sum of Distinct Positive nth Powers arxiv: v4 [math.nt] 9 Jul 2017 Doyon Kim Department of Mathematics and Statistics Auburn University Auburn, AL USA dzk0028@auburn.edu Abstract Itisknownthatforanarbitrarypositiveinteger nthesequencesx n = 1 n,2 n,... is complete, meaning that every sufficiently large integer is a sum of distinct nth powers of positive integers. We prove that every integer m b 12 n 1 r b 122n 1+2b 2 n 2a+ab, where a = n!2 n2, b = 2 n3 a n 1, r = 2 n2 n a, is a sum of distinct positive nth powers. 1 Introduction Let S = s 1,s 2,... be a sequence of integers. The sequence S is said to be complete if every sufficiently large integer can be represented as a sum of distinct elements of S. For a complete sequence S, the largest integer that is not representable as a sum of distinct elements of S is called the threshold of completeness of S. We let θ S denote the threshold of completeness of S. The threshold of completeness is often very difficult to find even for a simple sequence. For an arbitrary positive integer n, let Sx n denote the sequence of nth powers of positive integers, i.e., Sx n = 1 n,2 n,... The completeness of the sequence was proved in 1948, by Sprague [6]. In 1954, Roth and Szekeres [5] further generalized the result by proving that if fx is a polynomial that maps integers into integers, then Sf = f1,f2,... is complete if and only if fx has a positive leading coefficient and for any prime p there exists an integer m such that p does not divide fm. In 1964, Graham [2] re-proved the theorem of Roth and Szekeres using alternative elementary techniques. However, little is known about the threshold of completeness of Sx n. The value θ Sx n is known only for n 6. The values are as follows: θ Sx = 0, θ Sx 2 = 128 [7], θ Sx 3 = [2], θ Sx 4 = [3], θ Sx 5 = [4], θ Sx 6 = [1]. Sprague, Roth and Szekeres, and Graham proved that Sx n is complete, but they were not interested in the size of θ Sx n. The values θ Sx n for 3 n 6 were found by methods that require lengthy 1

2 calculations assisted by computer, and they do not give any idea on the size of θ Sx n for general n. In this paper, we establish an upper bound of θ Sx n as a function of n. Using the elementary techniques Graham used in his proof, it is possible to obtain an explicit upper bound of the threshold of completeness of Sx n = 1 n,2 n,3 n,... Since the case n = 1 is trivial, we let n be a positive integer greater than 1. We prove the following theorem: Theorem 1. Let a = n!2 n2, b = 2 n3 a n 1 and r = 2 n2 n a. Then θ Sx n < b 12 n 1 r b 122n 1+2b 2 n 2a+ab. The theorem yields the result θ Sx n = On! n n4 +n 3 +n 2 +2 ln3 ln2 n. The upper bound of θ Sx n given by the formula is much greater than 4 n4, while the actual values of θ Sx n for 2 n 6 are less than 4 n2. So the upper bound obtained in this paper is most likely far from being tight. 2 Preliminary results Let S = s 1,s 2,... be a sequence of integers. Definition 2. The set PS is a set of all sums of the form k=1 ǫ ks k where ǫ k is 0 or 1, all but a finite number of ǫ k are 0 and at least one of ǫ k is 1. Definition 3. The sequence S is complete if PS contains every sufficiently large integer. Definition 4. If S is complete, the threshold of completeness θ S is the largest integer that is not in PS. Definition 5. The set AS is a set of all sums of the form k=1 δ ks k where δ k is 1, 0 or 1 and all but a finite number of δ k are 0. Definition 6. Let k be a positive integer. The sequence S is a Σk-sequence if s 1 k, and n 1 s n k + s j, n 2. j=1 For example, if S = 2,4,8,16,... then S is a Σ2-sequence since 2 n = 2+ n 1 j=1 2j for all n 2. Definition 7. Let c and k be positive integers. The sequence S is c, k-representable if PS contains k consecutive integers c+j, 1 j k. 2

3 For example, if S = 1,3,6,10,... is a sequence of triangle numbers then S is 8,3- representable since {9,10,11} PS. Definition 8. For a positive integer m, we define Z m S to be the sequence α 1,α 2,..., where 0 α i < m and s i α i mod m for all i. The two following lemmas, slightly modified from Lemma 1 and Lemma 2 in Graham s paper [2], are used to obtain the upper bound. Lemma 9. For a positive integer k, let S = s 1,s 2,... be a strictly increasing Σk-sequence of positive integers and let T = t 1,t 2,... be c,k-representable. Then U = s 1,t 1,s 2,t 2,... is complete and θ U c. Proof. It suffices to prove that every positive integer greater than c belongs to PU. The proof proceeds by induction. Note that all the integers c+t, 1 t k belong to PT, and all the integers c+s 1 +t, 1 t k belong to PU. If 1 t k then c+t PT PU, and if k +1 t k +s 1, then 1 k s 1 +1 t s 1 k and we have Therefore all the integers c+t = c+t s 1 +s 1 PU. c+t, 1 t k +s 1 belong to PU. Now, let n 2 and suppose that all the integers n 1 c+t, 1 t k + belong to PU, and that for every such t there is a PU representation of c+t such that none of s m, m n is in the sum. Since all the integers c+t+s n, 1 t k+ n 1 j=1 s j belong to PU and c+1+s n c+1+k + n 1 j=1 s j, all the integers c+t, 1 t k + belong to PU. Since S is a strictly increasing sequence of positive integers, this completes the induction step and the proof of lemma. Lemma 10. Let S = s 1,s 2,... be a strictly increasing sequence of positive integers. If s k 2s k 1 for all k 2, then S is a Σs 1 -sequence. j=1 n j=1 s j s j 3

4 Proof. For k 2, we have Therefore, S is a Σs 1 -sequence. s k 2s k 1 = s k 1 +s k 1 s k 1 +2s k 2 = s k 1 +s k 2 +s k 2 s k 1 +s k 2 +2s k 3 k 1 s 1 + s j. j=1 Lemma 9 shows that if a sequence S can be partitioned into one Σk-sequence and one c,k-representable sequence then S is complete with θ S c. What we aim to do is to partition Sx n into two such sequences for some c and k. Let fx = x n and let Sf = f1,f2,... Let a = n!2 n2 and r = 2 n2 n a. Partition the elements of the sequence Sf into four sets B 1, B 2, B 3 and B 4 defined by so that and B 1 = {fαa+β : 0 α 2 n2 n 1, 1 β 2 n }, B 2 = {fαa+β : 0 α 2 n2 n 1, 2 n +1 β a, αa+β < 2 n2 n a}, B 3 = {f2 n2 n a,f2 n2 n a+2,f2 n2 n a+4,...}, B 4 = {f2 n2 n a+1,f2 n2 n a+3,f2 n2 n a+5,...}, B 1 B 2 = {f1,f2,...,fr 1} B 3 B 4 = {fr,fr+1,fr+2,...}. Let S, T, U and W be the strictly increasing sequences defined by S = s 1,s 2,...,s 2 n 2, s j B 1, T = t 1,t 2,..., t j B 3, U = u 1,u 2,..., u j B 1 B 3, W = w 1,w 2,..., w j B 2 B 4. Then the sequences U and W partition the sequence Sf. First, using Lemma 10, we show that W is a Σa-sequence. Lemma 11. For a = n!2 n2 and r = 2 n2 n a, fr +1 fr 1 < fa+2n +1 < f2n +2 fa f2 n

5 Proof. Re-write the inequalities as 1+ 2 r 1 n < 1+ 2n +1 a n < 1+ 1 n 2. 2 n +1 It is clear that r 1 > a 2 2 n +1 > 2n +1, which proves the first two inequalities. The proof of the third inequality 1+ 1 n n n 12 n is also straightforward. Corollary 12. The sequence W is a Σa-sequence. Proof. Note that w 1 = 2 n + 1 n. For every k 2, equalities: w k w k 1 satisfies one of the following w k = fα+1 w k 1 fα w k, for α 2 n +1; 1 = fβa+2n +1, for β 1; w k 1 fβa 2 w k fγ +2 =, for γ r 1. w k 1 fγ 3 Also, for every α 2 n +1, β 1 and γ r 1 we have fα+1 fα fβa+2 n +1 fβa fγ +2 fγ f2n +2 f2 n +1, fa+2n +1, fa fr+1 fr 1. w Thus, by Lemma 11, k w k 1 2 for k 2, and therefore by Lemma 10, W is a Σ2 n +1 n - sequence. To complete the proof, it remains to prove that 2 n +1 n < a for all n > 1. The inequality is true for n = 2 and n = 3, and for n > 3 we have Therefore, W is a Σa-sequence. 2 n +1 n < 2 n +2 n n = 2 n 2 n2 < n!2 n2 = a. 5

6 Now, we prove that U is d,a-representable for some positive integer d. By Lemma 9, the value d is the upper bound of θ Sx n. Note that the sequences S and T partition U. Lemma 13 shows that PS contains a complete residue system modulo a, and Lemma 14 and 15 together show that PT contains arbitrarily long arithmetic progression of integers with common difference a. The properties of S and T are used in Lemma 16 to prove that PU contains a consecutive integers. Lemma 13. The set PS contains a complete residue system modulo a. Proof. Itsufficestoprovethat{1,2,...,a} PZ a S. LetS 1, S 2,..., S 2 n bethesequences defined by S j = j n,j n,...,j n, 1 j 2 n where S j = 2 n2 n for all j. Since for each 0 α 2 n2 n 1, 1 β 2 n we have fαa+β β n mod a, and S is the sequence of such fαa+β in increasing order, the sequences S 1, S 2,..., S 2 n partition the sequence Z a S. Note that PS 1 = {1,2,...,2 n2 n }, PS 2 = {2 n,2 2 n,3 2 n,...,2 n2 n 2 n }. Since for every integer 1 m 2 n2 n 1+2 n there exist 0 α 2 n2 n, 1 β 2 n such that m = α2 n +β, we have Likewise, for every j 3, the inequality PS 1 S 2 = {1,2,3,...,2 n2 n 1+2 n }. j n < 2 n j 1 n < 2 n2 n 1+2 n + +j 1 n holds, and therefore for every 1 m 2 n2 n 1+2 n + +j n there exists 0 α 2 n2 n, 1 β 2 n2 n 1+2 n + j 1 n such that m = αj n +β. Therefore PZ a S = PS 1 S 2 S 2 n = {1,2,3,...,2 n2 n 1+2 n +3 n + +2 n2 }. It remains to prove that a = n!2 n2 2 n2 n 1+2 n +3 n + +2 n2. Since 1+2 n + +2 n2 2 n 1 n n 2 n, 6

7 we have 2 2 n2 n 1+2 n + +2 n n n n 2 n n +1 =. 2 Since 2 n +1 > 2j for every positive integer j n, we have 2 n2 n 1+2 n + +2 n2 2 n 2 n n +1 1 n!2 n2 2 n!2 n2 = 2n +1 n = n!2 n n 2 n +1 2j j=1 > 1. Therefore, a = n!2 n2 < 2 n2 n 1+2 n + +2 n2 and it completes the proof. Lemma 14. For every positive integer m, fm,fm+2,fm+4,...,fm+ 2 a A 3 22n 1. Proof. Define k : Q[x] Q[x] by: 1 gx = g4x+2 g4x, k gx = 1 k 1 gx, 2 k n, so that for 1 k n, k fx is a polynomial of degree n k. For example, and = 2 fx = 1 f4x+2 f4x f16x+10+f16x f16x+8+f16x+2 3 fx = 1 2 fx = f64x+42+f64x+32+f64x+8+f64x+2 f64x+40+f64x+34+f64x+10+f64x. It is easy to check that there are 2 k 1 positive terms and 2 k 1 negative terms in k fx, and all of the terms are distinct. Therefore, for each 1 k n, there exist 2 k distinct integers α k 1 > α k 2 > > α k 2 k 1, β k 1 > β k 2 > > β k 2 k 1 with α k 1 > β k 1 such that 2 k 1 k fx = f 2 2k x+α k i i=1 7 2 k 1 i=1 f 2 2k x+β k i.

8 Since α 1 1 = 2 and α k 1 = 4α k for k 2, we have α k 1 = k 1. Also, we have {α k 2 k 1,β k 2 k 1 } = {0,2}. Therefore f2 k fx A 2k x,f2 2k x+2,...,f2 2k x k 1. On the other hand, since we have Therefore, 1 fx = f4x+2 f4x = 4x+2 n 4x n = n2 2n 1 x n 1 +terms of lower degree, n fx = nn 1n n 1 2 2n 3 2 2n = n!2 n2 = a. f2 a A 2n x,f2 2n x+2,...,f2 2n x n 1. Since the n fx is a polynomial of degree 0, the value a = n fx is independent of x. Therefore, we can replace 2 2n x with an arbitrary positive integer m and we have fm,fm+2,fm+4,...,fm+ 2 a A 3 22n 1. Lemma 15. For every positive integer t, there exists a positive integer c such that all the integers c+ja, 1 j t belong to PT and c < t 12 n 1 r t 122n 1+2t 2 n a. Proof. Let α = n 1, and let T 1,T 2,...,T t 1 be the sequences defined by T 1 = fr,fr+2,fr+4,...,fr+α, T 2 = fr+α+2,fr+α+4,...,fr+2α+2, T 3 = fr+2α+4,fr+2α+6,...,fr+3α+4,... T t 1 = fr+t 2α+2t 2,...,fr+t 1α+2t 2. 8

9 By Lemma 14, a AT j for every 1 j t 1, and there exists A j,b j PT j such that A j B j = a, both A j and B j consist of 2 n 1 terms, and all 2 n terms of A j and B j are distinct. Let C 1 = B 1 +B 2 +B 3 + +B t 1, C 2 = A 1 +B 2 +B 3 + +B t 1, C 3 = A 1 +A 2 +B 3 + +B t 1,... j 1 t 1 C j = A i + B i,... i=1 i=j C t = A 1 +A 2 +A 3 + +A t 1. Then each C j belongs to PT, and C 1,C 2,...,C t is an arithmetic progression of t integers with common difference a. Thus, they are exactly the integers c + ja, 1 j t with c = C 1 a = B 1 +B 2 + +B t 1 a. Since each B j, 1 j t 1 is a sum of 2 n 1 terms in T, and all of the terms are less than or equal to fr+t 1α+2t 2 = r t 122n 1+2t 2 n, we have c = C 1 a < t 12 n 1 r t 122n 1+2t 2 n a. Finally, we show that PU contains a consecutive integers k 1 + t 1,k 2 + t 2,...,k a + t a, where {k 1,k 2,...,k a } is a complete residue system of a in PS and t 1,t 2,...,t a are taken from the arithmetic progression in PT. Lemma 16. Let b = 2 n3 a n 1. The sequence U is d,a-representable for a positive integer d such that d < b 12 n 1 r b 122n 1+2b 2 n 2a+ab. Proof. By Lemma 15, PT contains an arithmetic progression of b integers, c+ja, 1 j b with c < b 12 n 1 r b 122n 1+2b 2 n a. By Lemma 13, there exist positive integers 1 = k 1 < k 2 < < k a in PS such that {k 1,k 2,...,k a } is a complete residue system modulo a. For 1 j a, let ka k j n j = +1. a 9

10 Then for each 1 j a, k a k j a < n j k a k j a +1 k a < n j a+k j k a +a. Also, if i j then n i a+k i n j a+k j mod a. Therefore is the set of a consecutive integers {c+n 1 a+k 1,c+n 2 a+k 2,...,c+n a a+k a } {c+k a +1,c+k a +2,...,c+k a +a}. It remains to prove that each c + n j a + k j is in PU. Let ΣS denote the sum of every element of S. Since S = 2 n2, and s j f2 n2 n 1a+2 n = r a+2 n n < r n a 2 n n < r n n! for each s j S, we have Therefore, for each 1 j a we have ΣS < 2 n2 r n n! = 2 n2 r n a. 1 n j < k a a +1 1 a ΣS+1 < 1 a 2n2 r n = 2 n3 a n 1 = b and thus all of c +n j a belong to PT. Since all of k j belong to PS, all of c +n j a+k j belong to PU. Therefore, U is c+k a,a-representable. Let Since k a < ΣS < 2 n2 r n a = ab a, d = c+k a. d = c+k a < b 12 n 1 r+ 2 3 b 122n 1+2b 2 n 2a+ab. Now we have everything we need to prove the theorem. 3 Proof of the theorem Recall that U and W are disjoint subsequences of Sf. By Corollary 12, W is a Σa- sequence and by Lemma 16, U is d, a-representable with d < b 12 n 1 r b 122n 1+2b 2 n 2a+ab. Therefore by Lemma 9, Sx n = Sf is complete and θ Sx n d < b 12 n 1 r b 122n 1+2b 2 n 2a+ab. 10

11 4 Acknowledgments The author would like to thank Dr. Luke Oeding of Auburn University for his advice. His suggestions were valuable and helped the author to obtain a better upper bound. Also, the author would like to thank Dr. Peter Johnson of Auburn University and the anonymous referees for their helpful comments. References [1] C. Fuller and R. H. Nichols, Generalized Anti-Waring numbers, J. Integer Seq , Article [2] R. L. Graham, Complete sequences of polynomial values, Duke Math. J , [3] S. Lin, Computer experiments on sequences which form integral bases, in J. Leech, ed., Computational Problems in Abstract Algebra, Pergamon Press, 1970, [4] C. Patterson, The Derivation of a High Speed Sieve Device, Ph.D. thesis, University of Calgary, [5] K. F. Roth and G. Szekeres, Some asymptotic formulae in the theory of partitions, Q. J. Math , [6] R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z , [7] R. Sprague, Über Zerlegungen in ungleiche Quadratzahlen, Math. Z , Mathematics Subject Classification: Primary 11P05; Secondary 05A17. Keywords: complete sequence, threshold of completeness, sum of powers. Concerned with sequence A