Diophantine Problems and Powers

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E0123/: Asks whether, for pairwise coprime a, b, c above one, every large integer is a sum of distinct products of powers of a, b and c, none dividing another. Erdős also suggested the weaker hypothesis that a, b and c have no common factor, under which the answer is no, as 6, 10 and 15 show.

E0124/: Asks whether all large integers are sums of one number from each of several sets of sums of distinct powers of given bases satisfying a density condition.

E0125/: Asks whether the sumset of integers using only digits zero and one in base three and those using only those digits in base four has positive lower density.

E0137/: Asks whether the product of k consecutive positive integers, for some k at least three, can ever be powerful, meaning every prime dividing it divides it twice.

E0246/: Asks whether, for coprime a and b, every large integer is a sum of distinct numbers of the form a to the k times b to the l.

E0261/: Asks for which n the value n over two to the n is a sum of distinct terms k over two to the k, and whether some rational has uncountably many such sums.

E0322/: Estimates the number of ways an integer is a sum of k many kth powers, and asks whether it exceeds n to a fixed positive power infinitely often.

E0323/: Asks whether the integers up to x that are sums of k kth powers number at least x to the power 1 minus epsilon, and whether sums of m such powers, m below k, number at least a constant times x to the m over k.

E0324/: Asks whether some polynomial with integer coefficients has all sums of two of its values at distinct nonnegative integers distinct.

E0325/: Asks whether the number of integers up to x that are sums of three nonnegative kth powers is at least a constant times x to the power three over k.

E0363/: Asks whether only finitely many families of disjoint integer intervals, each of length at least four, have the product of all their members equal to a square.

E0364/: Asks whether three consecutive positive integers can all be powerful, meaning every prime dividing such a number divides it at least twice.

E0365/: Asks whether one of any two consecutive powerful numbers must be a square, and whether such pairs up to x number at most a power of the logarithm of x.

E0366/: Asks whether some integer divisible by the square of each of its prime factors is followed by one divisible by the cube of each; Erdős and Graham's companion question, with the cube first, is answered by 12167, 12168.

E0374/: Determines how many integers up to n need exactly k factorials, with the largest being that integer's, to form a square product, for k from three to six.

E0388/: Asks whether two disjoint blocks of more than three consecutive integers can have equal products only finitely often, and whether such cases can be classified.

E0389/: Asks whether every positive integer n admits some k for which the product of the first k integers from n divides the product of the next k.

E0394/: Bounds the average least starting point m for which n divides a product of k consecutive integers from m, and asks whether these averages shrink as k grows.

E0406/: Asks whether only finitely many powers of two are written with just the digits zero and one in base three.

E0407/: Asks whether the number of ways to write an integer as a power of two plus a power of three plus a product of a power of two and a power of three is bounded.

E0433/: Asks whether the largest non-representable integer for the worst coprime k-element subset of the first n integers is asymptotically n squared over k minus one.

E0434/: Determines which coprime k-element subset of the first n integers leaves the most integers unrepresentable as sums of its members, and whether it is the top k.

E0435/: The largest integer not expressible as a nonnegative integer combination of the binomial coefficients C(n,1), ..., C(n,n-1), for n not a prime power.

E0437/: Estimates how many of the partial products of an increasing sequence bounded by x can be perfect squares, and whether nearly x of them can be.

E0443/: How many values are shared by the two sets of products k times m minus k and l times n minus l for m different from n, and whether this count can be large or must be very small.

E0445/: Asks whether, for any exponent above one half and any large prime, every interval of that length contains two numbers whose product is one modulo the prime.

E0477/: Asks whether the image of an integer polynomial of degree at least two admits a unique additive complement in the integers.

E0479/: Asks whether, for every k other than one, there are infinitely many n with two to the n congruent to k modulo n.

E0493/: Asks whether there is a fixed k such that every large enough integer is the product of k integers at least two minus their sum.

E0672/: Asks whether a product of at least four positive terms in a primitive arithmetic progression, with gcd of initial term and difference one, can be a perfect power.

E0674/: Asks whether x to the power x times y to the power y equals z to the power z has integer solutions with x, y and z all greater than one.

E0676/: Asks whether every sufficiently large integer has the form a times a prime squared plus b, with a at least one and b less than that prime.

E0686/: Asks whether every integer at least 2 is a ratio of two products of k consecutive integers, for some k at least 2 with the blocks disjoint.

E0782/: Asks whether the squares contain arbitrarily long progressions with bounded gap error, and arbitrarily large sets of all zero-one sums of given numbers.

E0829/: Asks whether the number of ways to write n as a sum of two cubes is at most a power of the logarithm of n.

E0841/: Estimates the least length of a run of integers just above n that contains a subset whose product with n is a perfect square; Erdős's original question, whether the n with t_n at least n^{1-o(1)} have density zero, has the answer yes.

E0843/: Asks whether the squares are Ramsey 2-complete, so that any 2-coloring of them leaves all large integers as sums of distinct same-colored squares.

E0845/: Asks whether, for each constant C, the sums of distinct products of powers of two and three lying within a factor C of each other have density zero.

E0913/: Asks whether infinitely many n have all exponents distinct in the prime factorization of n times n plus one.

E0930/: Asks whether, for every r, some k makes the product of all integers in any r disjoint intervals of length at least k never a perfect power.

E0931/: Asks whether only finitely many disjoint blocks of consecutive integers, of lengths k1 and k2 at least 3, have products with the same prime factors.

E0933/: Asks whether the largest divisor of n times n plus 1 built only from the primes 2 and 3 exceeds any fixed multiple of n log n for suitable n.

E0935/: Asks whether the powerful part of n(n+1)...(n+l) is below n^(2+eps) for every eps once n is large, whether its ratio to n squared is unbounded when l is at least 2, and whether its ratio to n^(l+1) tends to zero.

E0936/: Asks whether 2 to the n plus or minus 1 and n factorial plus or minus 1 are powerful numbers for only finitely many n.

E0937/: Asks whether there are infinitely many four-term arithmetic progressions made of pairwise coprime powerful numbers.

E0938/: Concerns the increasing sequence of powerful numbers, those integers divisible by the square of every prime dividing them, and the gaps between them.

E0939/: Concerns sums of coprime r-powerful numbers; the 3-powerful triple question is answered yes (Nitaj 1995), infinitely many solutions exist for every r at least 6, and r = 4 is open.

E0940/: Concerns r-powerful numbers for r at least 3, the integers divisible by the r-th power of each of their prime factors.

E0941/: Asks whether every sufficiently large integer is the sum of at most three powerful numbers.

E0942/: Estimates how many powerful integers lie between consecutive squares, in particular whether some fixed power of log n bounds that count for every n and is nearly reached for infinitely many n.

E0943/: Asks whether the number of ways of writing n as a sum of two powerful numbers is smaller than any fixed power of n.

E0969/: Determines the order of magnitude of the error term when the count of squarefree integers up to x is compared with six over pi squared times x.

E0978/: Concerns the values of an irreducible integer polynomial with positive leading coefficient whose degree exceeds two and is not a power of two.

E0979/: Asks whether, for each k at least two, the number of ways to write n as a sum of k kth powers of primes is unbounded.

E1056/: Asks whether, for every k at least two, there are a prime and k consecutive intervals of integers whose products are each congruent to one modulo that prime.

E1058/: Asks whether only finitely many n lying between two consecutive primes have n factorial plus one divisible only by the next two primes.

E1065/: Asks whether infinitely many primes are one plus a power of two times a prime, or one plus a power of two times a power of three times a prime.

E1072/: Studies the least n for which a given prime divides n factorial plus one, as a function of that prime.

E1081/: Asks whether the count of integers up to x that are sums of two squarefull numbers is asymptotic to a constant times x over the square root of log x.

E1107/: Asks whether every large integer is the sum of at most r plus one numbers divisible by the r-th power of each of their prime factors, for r at least 2.

E1110/: Concerns which integers are sums of numbers of the form a power of p times a power of q, none dividing another, for coprime integers p greater than q.

E1140/: Asks whether infinitely many n make n - 2x^2 prime for every x with 2x^2 < n.

E1148/: Asks whether every large integer is a sum of two squares minus a square with all three squares at most that integer.

E1214/: Asks whether two positive integers x and y must be equal when the primes dividing x to the n minus one match those dividing y to the n minus one for every n.

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Problems asking whether an equation has integer solutions or whether integers of a given algebraic shape exist — squares and perfect powers, powerful and squarefree numbers, sums of k-th powers, products of consecutive integers, and base and digit representations.

Site tags routed here: base representations, complete sequences, number theory, powerful, powers, sidon sets.