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Text A set A of positive integers is called d-complete if every sufficiently large integer is the sum of distinct terms taken from A such that no one divides the other. In this paper we answer two questions of Erdős and Lewin partially and settle a conjecture of Erdős and Lewin on d-complete sequences affirmatively. We also pose two conjectures for further research. Video For a video summary of this paper, please visit https://www.youtube.com/watch?v=I0AoLpvj4QQ .

For any prime p, let R p be the set of all quadratic residues modulo p. In 2012, Sarkozy proved that if p is a sufficiently large prime, then R p has no 3-decomposition A + B + C = R p with | A | , | B | , | C | ≥ 2 . In this paper, we prove that for any prime p, R p has no additive 3-decomposition A + B + C = R p with | A | , | B | , | C | ≥ 2 . Furthermore, for any prime p, if A + B = R p is a 2-decomposition, then 0.17 p + 1 | A | , | B | 2.8 p − 6.63 . We also pose three related conjectures.

Text In 1988, Erdős, Nathanson and Sarkozy proved that, if A is a set of nonnegative integers with the lower asymptotic density 1 / k , where k is a positive integer, then ( k + 1 ) A must contain an infinite arithmetic progression with difference at most k 2 − k , where ( k + 1 ) A is the set of all sums of k + 1 elements of A, and asked if ( k + 1 ) A must contain an infinite arithmetic progression with difference at most O ( k ) . In this paper, we solve this problem by proving that, for every sufficiently large integer k, there exists a set A of nonnegative integers with the lower asymptotic density 1 / k such that ( k + 1 ) A does not contain an infinite arithmetic progression with difference less than k 1.5 . We also study the similar problem when A has the lower asymptotic density 0 w 1 . Video For a video summary of this paper, please visit https://youtu.be/B1QJc2IFSIw .

Text Let v n be the denominator of the n-th harmonic number H n = 1 + 2 − 1 + ⋯ + n − 1 . In this paper, we prove the following unusual result: the set of positive integers n with v n = v n + 1 has density one. We also prove that the same result is true for the denominators of alternating harmonic numbers. Video For a video summary of this paper, please visit https://youtu.be/Sk0zMSKPe3M .

Text For each positive integer n, let L ( n ) be the length of longest arithmetic progressions in the least positive reduced residue system modulo n. In this paper, we give an asymptotic formula for ∑ n ≤ x L ( n ) . This solves a problem of Pongsriiam (2018) [4] . Video For a video summary of this paper, please visit https://youtu.be/xJEpSBYoZJ0 .

Text For any positive real number r, let p r ( n ) be the number of solutions of the equation n = ⌊ a 1 r ⌋ + ⋯ + ⌊ a k r ⌋ with integers 1 ≤ a 1 ≤ ⋯ ≤ a k . Recently, Luca and Ralaivaosaona gave an asymptotic formula for p 2 ( n ) . In Part I, it is proved that, for any real number r > 1 , we have exp ⁡ ( τ 1 n r / ( r + 1 ) ) ≤ p r ( n ) ≤ exp ⁡ ( τ 2 n r / ( r + 1 ) ) for two positive constants τ 1 and τ 2 (depending only on r). In this paper, we prove that, for any real number r > 1 , p r ( n ) = exp ⁡ ( c 1 n r / ( r + 1 ) + c 2 n ( r − 1 ) / ( r + 1 ) + ⋯ + c l n ( r − l + 1 ) / ( r + 1 ) + O ( n 1 / ( r + 1 ) ) ) , where l is the integer with l r ≤ l + 1 and c 1 , c 2 , … , c l are computable constants depending only on r. In particular, c 1 = ( 1 + r − 1 ) ( r ζ ( r + 1 ) Γ ( r + 1 ) ) 1 / ( r + 1 ) . For 0 r 1 , we pose a conjecture for the asymptotic formula for p r ( n ) . Video For a video summary of this paper, please visit https://youtu.be/Z9HiSbr9eJY .

Text A set A of nonnegative integers is said to be complete if every sufficiently large natural number is the sum of distinct terms taken from A. In 1959, Birch confirmed a conjecture of Erdős by proving that the set { p n q m : n , m = 0 , 1 , … } is complete, where p and q are two coprime integers greater than 1. In this paper, we study extensively the set P ( S p 1 , … , p r ) of all vectors ( m 1 , … , m r ) with integer coordinates which can be represented as ∑ ( p 1 α 1 , … , p r α r ) , where p 1 , … , p r are integers greater than 1 and α 1 , … , α r are nonnegative integers. We find many regular parts in P ( S p 1 , … , p r ) , for example, discrete rectangles. To study the problem, we find a new matching principle. Video For a video summary of this paper, please visit https://youtu.be/qXUe1E4jlLQ .

Text A positive integer n is called weakly prime-additive if n has at least two distinct prime divisors and there exist distinct prime divisors p 1 , … , p t of n and positive integers α 1 , … , α t such that n = p 1 α 1 + ⋯ + p t α t . It is clear that t ≥ 3 . In 1992, Erdős and Hegyvari proved that, for any prime p, there exist infinitely many weakly prime-additive numbers with t = 3 which are divisible by p. In this paper, we prove that, for any positive integer m, there exist infinitely many weakly prime-additive numbers with t = 3 which are divisible by m if and only if 8 ∤ m . We also present some related results and pose several problems for further research. Video For a video summary of this paper, please visit https://youtu.be/WC_VRFtY07c .

Text Two infinite sequences A and B of nonnegative integers are called additive complements, if their sum contains all sufficiently large integers. We also say that B is an additive complement of A if A and B are additive complements. In this paper, we consider a problem of Ben Green on additive complements of the squares: S = { 1 2 , 2 2 , … } . The following result is proved: if B = { b n } n = 1 ∞ with b n ≥ π 2 16 n 2 − 0.57 n 1 2 log ⁡ n − β n 1 2 for all positive integers n and any given constant β, then B is not an additive complement of S. In particular, B = { ⌊ π 2 16 n 2 ⌋ | n = 1 , 2 , … } is not an additive complement of S. Video For a video summary of this paper, please visit https://youtu.be/cVXWCP4Igp8 .

Text Let H n be the n-th harmonic number and let u n be its numerator. For any prime p, let J p be the set of positive integers n with p | u n . In 1991, Eswarathasan and Levine conjectured that J p is finite for any prime p. It is clear that the p-adic valuation of H n is not less than − ⌊ log p ⁡ n ⌋ . Let T p be the set of positive integers n such that the p-adic valuation of H n is equal to − ⌊ log p ⁡ n ⌋ . Recently, Carlo Sanna proved that | J p ∩ [ 1 , x ] | 129 p 2 / 3 x 0.765 and that there exists S p ⊆ T p with δ ( S p ) > 0.273 , where δ ( X ) denotes the logarithmic density of the set X of positive integers. He also commented that with his methods δ ( S p ) > 1 / 3 − e cannot be achieved. In this paper, we improve these results. For example, two of our results are: (a) | J p ∩ [ 1 , x ] | ≤ 3 x 2 / 3 + 1 / ( 25 log ⁡ p ) ; (b) δ ( T p ) exists and 1 − ( 2 log ⁡ p ) − 1 ≤ δ ( T p ) ≤ 1 − ( p log ⁡ p ) − 1 for all primes p ≥ 13 . In particular, δ ( T p ) > 0.63 for all primes p. Video For a video summary of this paper, please visit https://youtu.be/3ujCuVwH8k8 .

Text In 2014, Herzog, Kaplan and Lev introduced the critical numbers cr ( r , n ) of natural intervals [ r , n ] and determined the values of cr ( 1 , n ) and cr ( 2 , n ) for all n, and for r ≥ 3 , they determined the values of cr ( r , n ) for n ≤ ( 3 r 2 + 3 r ) / 2 and n ≥ ( 5 r 2 + r ) / 2 . In this note, we determine the remaining values of cr ( r , n ) for r ≥ 3 . Video For a video summary of this paper, please visit https://youtu.be/hZSI-Ul3K94 .

Text Let f ( n ) be a multiplicative function such that f does not vanish at some prime p 0 . In this paper, it is proved that, for any given integer n 0 with 1 ≤ n 0 ≤ 10 6 , if f ( p + q + n 0 ) = f ( p ) + f ( q ) + f ( n 0 ) for all primes p and q, then f must be the identity function: f ( n ) = n for all integers n ≥ 1 . If a variation of Goldbach's conjecture is true, then the result is true for any fixed integer n 0 . Video For a video summary of this paper, please visit https://youtu.be/uGtHiVBZGj0 .

Text A set A of positive integers is called d-complete if every sufficiently large integer is the sum of distinct terms taken from A such that no one divides the other. In 1996, Erdős and Lewin proved that the sequence { 2 α 5 β p γ } is d-complete for every prime p with 6 p 20 and commented that they cannot prove that the sequence { 2 α 5 β 23 γ } is d-complete. In this paper, we prove that, for any integer p > 6 with ( p , 10 ) = 1 , if there exists a number c = c ( p ) such that every integer n with c n 25 p c is the sum of distinct terms taken from { 2 α 5 β p γ } such that no one divides the other, then the sequence { 2 α 5 β p γ } is d-complete. In particular, we prove that the sequence { 2 α 5 β p γ } is d-complete for p ∈ { 9 , 21 , 23 , 27 , 29 , 31 } . Video For a video summary of this paper, please visit https://youtu.be/lq7XleRNLf8 .

For a positive integer n, let σ(n) and γ(n) denote the sum of divisors and the product of distinct prime divisors of n, respectively. It is known that, if σ(n)=γ(n)2, then at most two exponents of odd primes are equal to 1 in the prime factorization of n. In this paper, we prove that, if σ(n)=γ(n)2 and only one exponent is equal to 1 in the prime factorization of n, then (1) n is divisible by 3; (2) n is divisible by the fourth powers of at least two odd primes; (3) at least two exponents of odd primes are equal to 2. We also prove that, if σ(n)=γ(n)2, then at least half of the exponents α of the primes have the property that the numbers α+1 must be either primes or prime squares.

Text In this paper, the following results are proved: (i) For any odd integer l with at most two distinct prime factors and any positive integer n , the product (1l+1)(2l+1)⋯(nl+1)(1l+1)(2l+1)⋯(nl+1) is not a powerful number; (ii) For any integer r≥1r≥1, there exists a positive integer TrTr such that, if l is a positive odd integer with at most r distinct prime factors and n is an integer with n≥Trn≥Tr, then (1l+1)(2l+1)⋯(nl+1)(1l+1)(2l+1)⋯(nl+1) is not a powerful number. Video For a video summary of this paper, please visit http://youtu.be/nU-nkxNX1BA.

Text For any positive integer n, let n = q 1 ⋯ q s be the prime factorization of n with q 1 ≥ ⋯ ≥ q s > 1 . A positive integer n is said to be ordinary if the smallest positive integer with exactly n divisors is p 1 q 1 − 1 ⋯ p s q s − 1 , where p k denotes the kth prime. Let ⌊ x ⌋ be the largest integer not exceeding x. In 2006, Brown proved that all square-free integers are ordinary and the set of all ordinary integers has asymptotic density one. In this paper, we prove that, if q ⌊ s ⌋ ≥ 9 ( log s ) 2 , then n is ordinary. Furthermore, the set of such integers n has asymptotic density one. We also determine all ordinary integers which are not divisible by any fifth power of a prime. Video For a video summary of this paper, please visit http://youtu.be/UeIMWjRFUnA .

Text Two infinite sequences A and B of non-negative integers are called additive complements, if their sum contains all sufficiently large integers. Let A ( x ) and B ( x ) be the counting functions of A and B. In 1994, Sarkozy and Szemeredi proved that, for additive complements A and B, if lim sup A ( x ) B ( x ) / x ⩽ 1 , then A ( x ) B ( x ) − x → + ∞ as x → + ∞ . In 2010, the authors generalized this result and proved that if lim sup A ( x ) B ( x ) / x 5 / 4 or lim sup A ( x ) B ( x ) / x > 2 , then A ( x ) B ( x ) − x → + ∞ as x → + ∞ . In 2011, the authors pointed out that the constant 2 cannot be improved. In this paper, we improve 5/4 to 3 − 3 . Video For a video summary of this paper, please click here or visit http://youtu.be/fSTUWhPJdtw .

Text For a sequence A of nonnegative integers, let P ( A ) be the set of all integers which can be represented as the finite sum of distinct terms of A and for the empty sum will be 0. A sequence A of nonnegative integers is called complete if P ( A ) contains all sufficiently large integers. For a sequence S = { s 1 , s 2 , … } of positive integers and a positive real number α, let S α denote the sequence { ⌊ α s 1 ⌋ , ⌊ α s 2 ⌋ , … } , where ⌊ x ⌋ denotes the greatest integer not greater than x. Let U S = { α | S α is complete } . In 1995, Hegyvari proved the following theorem: If lim n → ∞ ( s n + 1 − s n ) = + ∞ , s n + 1 γ s n for all integers n ⩾ n 0 , where 1 γ 2 , and U S ≠ ∅ , then μ ( U S ) > 0 , where μ ( U S ) is the Lebesgue measure of U S . In this paper, we remove the condition lim ( s n + 1 − s n ) = + ∞ . Furthermore, we prove that, if s n + 1 γ s n for all integers n ⩾ n 0 , where 1 γ ⩽ 7 / 4 , then μ ( U S ) > 0 . We also pose a problem for further research. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=PXQ9e5hw3Hw .

Text In this paper, the following results are proved: (i) For any odd integer l with at most two distinct prime factors and any positive integer n, the product ( 1 l + 1 ) ( 2 l + 1 ) ⋯ ( n l + 1 ) is not a powerful number; (ii) For any integer r ≥ 1 , there exists a positive integer T r such that, if l is a positive odd integer with at most r distinct prime factors and n is an integer with n ≥ T r , then ( 1 l + 1 ) ( 2 l + 1 ) ⋯ ( n l + 1 ) is not a powerful number. Video For a video summary of this paper, please visit http://youtu.be/nU-nkxNX1BA .

A triple (a,b,c) ( a , b , c ) of positive integers is called a Markoff triple if it satisfies the Diophantine equation a 2 +b 2 +c 2 =3abc a 2 + b 2 + c 2 = 3 a b c . A famous old conjecture says that any Markoff triple (a,b,c) ( a , b , c ) with a⩽b⩽c a ⩽ b ⩽ c is determined uniquely by its largest member c . Let (a,b,c) ( a , b , c ) be a Markoff triple with a⩽b⩽c a ⩽ b ⩽ c . In 2001, Button proved that if c is of the form kp l k p l , where k is an integer with 1⩽k⩽10 35 1 ⩽ k ⩽ 10 35 and p l p l is a prime power, then c uniquely determines a and b . In this paper, as a complement to the result of Button, we prove that if either 3c−2 3 c − 2 or 3c+2 3 c + 2 is of the form kp l k p l , where k is an integer with 1⩽k⩽10 10 1 ⩽ k ⩽ 10 10 and p l p l is a prime power, then c uniquely determines a and b .

Text For any given two positive integers k 1 and k 2 , and any set A of nonnegative integers, let r k 1 , k 2 ( A , n ) denote the number of solutions of the equation n = k 1 a 1 + k 2 a 2 with a 1 , a 2 ∈ A . In this paper, we determine all pairs k 1 , k 2 of positive integers for which there exists a set A ⊆ N such that r k 1 , k 2 ( A , n ) = r k 1 , k 2 ( N ∖ A , n ) for all n ⩾ n 0 . We also pose several problems for further research. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=EnezEsJl0OY .

In this paper, we determine all sets A of integers such that, for any integral-valued polynomial f ( x ) which has no fixed divisor, for all integers l ⩾ 1 and n, there are infinitely many integers m > l and a choice of e i ∈ A such that n = e l f ( l ) + e l + 1 f ( l + 1 ) + ⋯ + e m f ( m ) . The earlier result shows that A = { − 1 , 1 } is such a set.

Text Let A 3 be the set of all positive integers pqr, where p, q, r are primes such that at least two of them are not equal. Denote by P ( n ) the largest prime factor of n. For n = p q r ∈ A 3 , define w ( n ) : = P ( p + q ) P ( p + r ) P ( q + r ) . In 2006, Wushi Goldring proved that for any n ∈ A 3 , there exists an i such that w i ( n ) ∈ { 20 , 98 , 63 , 75 } , where w 0 ( n ) = n and w i ( n ) = w ( w i − 1 ( n ) ) ( i ⩾ 1 ). If w ( m ) = n , then m is called a parent of n. Let B 3 be the set of all positive integers p q 2 of A 3 . In this paper, we study the function w extensively. For example, one of our results is that there exist infinitely many n ∈ B 3 which have at least n 1.1886 parents in B 3 . Several open problems are posed. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=8FWvR8_KoHA .