"property that when represented as a sum of two prime numbers (per Goldbach's conjecture) both of the primes must be greater than or equal to 23. There are exactly 220 different ways of partitioning 64 = 8 into a sum of square numbers. It is a tetrahedral number, the sum of the first ten triangular numbers, and a dodecahedral number. If all of the diagonals of a regular decagon are drawn, the resulting figure will have exactly 220 regions. It is the sum of the sums of the divisors of the first 16 positive integers. The number 220 can also"
"41 + 43 + 47), Motzkin number, self number. A Lucas and Fibonacci pseudoprime. ""See 323 (disambiguation)"" 324 = 2 × 3 = 18. 324 is the sum of four consecutive primes (73 + 79 + 83 + 89), totient sum of the first 32 integers, untouchable number, and a Harshad number. 325 = 5 × 13. 325 is a triangular number, hexagonal number, nonagonal number, centered nonagonal number. 325 is the smallest number to be the sum of two squares in 3 different ways: 1 + 18, 6 + 17 and 10 + 15. 325 is also the smallest"
"because . For example, One number less than a square (""m"" - 1) is always the product of - 1 and + 1 (e.g. 8 × 6 equals 48, while 7 equals 49). Thus, 3 is the only prime number one less than a square. A square number is also the sum of two consecutive triangular numbers. The sum of two consecutive square numbers is a centered square number. Every odd square is also a centered octagonal number. Another property of a square number is that (except 0) it has an odd number of positive divisors, while other natural numbers"
"1 to 6: there is no smaller number divisible by the numbers 1 to 5. It is the smallest number with exactly 12 divisors. It is one of seven integers that have more divisors than any number less than twice itself , one of six that are also lowest common multiple of a consecutive set of integers from 1, and one of six that are divisors of every highly composite number higher than itself. It is the sum of a pair of twin primes (29 + 31) and the sum of four consecutive primes (11 + 13 + 17 +"
"not enough to qualify for the next class. <br>The smallest normal simple magic cube is order 3. Minimum correct summations required = 3""m"" + 4 Each of the 3""m"" planar arrays must be a simple magic square. The 6 oblique squares are also simple magic. The smallest normal diagonal magic cube is order 5.<br> These squares were referred to as ‘Perfect’ by Gardner and others! At the same time he referred to Langman’s 1962 pandiagonal cube also as ‘Perfect’.<br> Christian Boyer and Walter Trump now consider this ""and"" the next two classes to be ""Perfect"". (See ""Alternate Perfect"" below).<br>A. H."
"not equal to 216 can be written in the form where is or a prime, and is a triangular number. In base 10, it is a Harshad number. There are 216 fixed hexominoes, the polyominoes made from 6 squares. 216 is a Friedman number. 216 is the smallest number ""n"", for which ""n""−3, ""n""−2, ""n""−1, ""n""+1, ""n""+2, ""n""+3 are all semiprimes. 216 (number) 216 (two hundred [and] sixteen) is the natural number following 215 and preceding 217. Since it is the smallest cube that is also the sum of three cubes (Plato was among the first to notice this, and"
"216 (number) 216 (two hundred [and] sixteen) is the natural number following 215 and preceding 217. Since it is the smallest cube that is also the sum of three cubes (Plato was among the first to notice this, and mentioned it in Book VIII of ""Republic""). It is also the sum of a twin prime (107 + 109). But since there is no way to express it as the sum of the proper divisors of any other integer, it is an untouchable number. This multiplicative magic square formula_1 has magic constant 216. It has been conjectured that each natural number"
"271 (number) 271 (two hundred [and] seventy-one) is the natural number after and before . 271 is a twin prime with 269, a cuban prime (a prime number that is the difference of two consecutive cubes), and a centered hexagonal number. It is the smallest prime number bracketed on both sides by numbers divisible by cubes, and the smallest prime number bracketed by numbers with five primes (counting repetitions) in their factorizations: After 7, 271 is the second-smallest Eisenstein–Mersenne prime, one of the analogues of the Mersenne primes in the Eisenstein integers. 271 is the largest prime factor of the"
"simple way as a subquotient. Twenty-six is a repdigit in base three (222) and in base twelve (22). In base ten, 26 is the smallest number that is not a palindrome to have a square (26 = 676) that is a palindrome. Twenty-six is: 26 (number) 26 (twenty-six) is the natural number following 25 and preceding 27. 26 is the only integer that is one greater than a square (5 + 1) and one less than a cube (3 − 1). A rhombicuboctahedron has twenty-six faces. When a 3 × 3 × 3 cube is made of twenty-seven unit cubes,"
"are integers slightly over 1024. 1024×768 pixels and 1280×1024 pixels are common standards of display resolution. 1024 (number) 1024 is the natural number following 1023 and preceding 1025. 1024 is a power of two: formula_1 (2 to the 10th power). It is the lowest power of two requiring four decimal digits, and the lowest power of two containing the digit 0 in its decimal representation (excluding any leading zeroes). It is also the square of 32: formula_2. 1024 is the smallest number with ""exactly"" 11 divisors (but note that there are smaller numbers with more than 11 divisors; e.g., 60"
"26 (number) 26 (twenty-six) is the natural number following 25 and preceding 27. 26 is the only integer that is one greater than a square (5 + 1) and one less than a cube (3 − 1). A rhombicuboctahedron has twenty-six faces. When a 3 × 3 × 3 cube is made of twenty-seven unit cubes, twenty-six of them are viewable as the exterior layer. There are 26 sporadic groups. The 26-dimensional Lorentzian unimodular lattice II plays a significant role in sphere packing problems and the classification of finite simple groups. In particular, the Leech lattice is obtained in a"
"25 (number) 25 (twenty-five) is the natural number following 24 and preceding 26. It is a square number, being 5 = 5 × 5. It is one of two two-digit numbers whose square and higher powers of the number also ends in the same last two digits, e.g. 25 = 625, the other is 76. It is the smallest square that is also a sum of two (non-zero) squares: 25 = 3 + 4. Hence it often appears in illustrations of the Pythagorean theorem. 25 is the sum of the single-digit odd natural numbers 1, 3, 5, 7 and 9,"
"888 (number) 888 (eight hundred [and] eighty-eight) is the natural number following 887 and preceding 889. 888 is a repdigit (a number all of whose digits are equal), and a strobogrammatic number (one that reads the same upside-down on a seven-segment calculator display). 888 = 700227072 is the smallest cube in which each digit occurs exactly three times, and the only cube in which three distinct digits each occur three times. 888 the smallest multiple of 24 whose digit sum is 24, and as well as being divisible by its digit sum it is divisible by all of its digits."
"this rule as outdated. Twelve is a composite number, the smallest number with exactly six divisors, its divisors being 1, 2, 3, 4, 6 and 12. Twelve is also a highly composite number, the next one being twenty-four. Twelve is the smallest abundant number, since it is the smallest integer for which the sum of its proper divisors (1 + 2 + 3 + 4 + 6 = 16) is greater than itself. Twelve is a sublime number, a number that has a perfect number of divisors, and the sum of its divisors is also a perfect number. Since there"
"1024 (number) 1024 is the natural number following 1023 and preceding 1025. 1024 is a power of two: formula_1 (2 to the 10th power). It is the lowest power of two requiring four decimal digits, and the lowest power of two containing the digit 0 in its decimal representation (excluding any leading zeroes). It is also the square of 32: formula_2. 1024 is the smallest number with ""exactly"" 11 divisors (but note that there are smaller numbers with more than 11 divisors; e.g., 60 has 12 divisors) . The neat coincidence that 2 is nearly equal to 10 provides the"
"if and only if one can arrange solid unit cubes into a larger, solid cube. For example, 27 small cubes can be arranged into one larger one with the appearance of a Rubik's Cube, since . The difference between the cubes of consecutive integers can be expressed as follows: or There is no minimum perfect cube, since the cube of a negative integer is negative. For example, . Unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25, 75 and 00 can be"
"too many times this way. This reduces the number of positions by 12^2. The final count is (24!)/72=8,617,338,912,961,658,880,000 total possible positions. Square-1 (puzzle) The Square-1, also known as Back to Square One and Cube 21, is a puzzle similar to the Rubik's Cube. Its distinguishing feature among the numerous Rubik's Cube variants is that it can change shape as it is twisted, due to the way it is cut, thus adding an extra level of challenge and difficulty. The Super Square One and Square Two puzzles have been recently introduced. The Super Square One has two additional layers that can"
"Taxicab number In mathematics, the ""n""th taxicab number, typically denoted Ta(""n"") or Taxicab(""n""), also called the ""n""th Hardy–Ramanujan number, is defined as the smallest number that can be expressed as a sum of two ""positive"" cube numbers in ""n"" distinct ways. The most famous taxicab number is 1729 = Ta(2) = 1 + 12 = 9 + 10. The name is derived from a conversation in about 1919 involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy: The concept was first mentioned in 1657 by Bernard Frénicle de Bessy, and was made famous in the early 20th"
"five-digit repunit 11111, and the largest prime number for which the decimal period of its multiplicative inverse is 5: 271 (number) 271 (two hundred [and] seventy-one) is the natural number after and before . 271 is a twin prime with 269, a cuban prime (a prime number that is the difference of two consecutive cubes), and a centered hexagonal number. It is the smallest prime number bracketed on both sides by numbers divisible by cubes, and the smallest prime number bracketed by numbers with five primes (counting repetitions) in their factorizations: After 7, 271 is the second-smallest Eisenstein–Mersenne prime, one"
"square is less than 2 is a set with an upper bound (e.g. 1.5) but no (rational) least upper bound: hence the rational numbers do not satisfy the least upper bound property. A main reason for using real numbers is that the reals contain all limits. More precisely, a sequence of real numbers has a limit, which is a real number, if (and only if) its elements eventually come and remain arbitrarily close to each other. This is formally defined in the following, and means that the reals are complete (in the sense of metric spaces or uniform spaces, which"
"24 squares; however, it was not until 1982 that Duijvestijn, Pasquale Joseph Federico and P. Leeuw mathematically proved it to be the lowest-order example. When the constraint of all the squares being different sizes is relaxed, a squared square such that the side lengths of the smaller squares do not have a common divisor larger than 1 is called a ""Mrs. Perkins's quilt"". In other words, the greatest common divisor of all the smaller side lengths should be 1. The Mrs. Perkins's quilt problem is to find a Mrs. Perkins's quilt with the fewest pieces for a given square. A"
"4 4 (four) is a number, numeral, and glyph. It is the natural number following 3 and preceding 5. Four is the smallest composite number, its proper divisors being and . 4 is the smallest squared prime (""p"") and the only even number in this form. 4 is also the only square one more than a prime number. A number is a multiple of 4 if its last two digits are a multiple of 4. For example, 1092 is a multiple of 4 because . In addition, 2 + 2 = 2 × 2 = 2 = 4. Continuing the"
"50 (number) 50 (fifty) is the natural number following 49 and preceding 51. Fifty is the smallest number that is the sum of two non-zero square numbers in two distinct ways: 50 = 1 + 7 = 5 + 5. It is also the sum of three squares, 50 = 3 + 4 + 5, and the sum of four squares, 50 = 6 + 3 + 2 + 1. It is a Harshad number. There is no solution to the equation φ(""x"") = 50, making 50 a nontotient. Nor is there a solution to the equation ""x"" − φ(""x"")"
"at most formula_8, or formula_9. For formula_10 and above, so any number over 1000 gets smaller under this process and in particular becomes a number with strictly fewer digits. Once we are under 1000, the number for which the sum of squares of digits is largest is 999, and the result is 3 times 81, that is, 243. Considering more precisely the intervals [244, 999], [164, 243], [108, 163] and [100, 107], we see that every number above 99 gets strictly smaller under this process. Thus, no matter what number we start with, we eventually drop below 100. An exhaustive"
"using the separation of variables technique. Seventeen is the sixth Mersenne prime exponent, yielding 131071. Seventeen is the first number that can be written as the sum of a positive cube and a positive square in two different ways; that is, the smallest ""n"" such that ""x"" + ""y"" = ""n"" has two different solutions for ""x"" and ""y"" positive integers. The next such number is 65. Seventeen is the minimum number of vertices on a graph such that, if the edges are coloured with three different colours, there is bound to be a monochromatic triangle. (See .) Seventeen is"
"the least positive integer ""s"" such that every sufficiently large integer (i.e. every integer greater than some constant) can be represented as a sum of at most ""s"" positive integers to the power of ""k"". Clearly, ""G""(1) = 1. Since squares are congruent to 0, 1, or 4 (mod 8), no integer congruent to 7 (mod 8) can be represented as a sum of three squares, implying that . Since for all ""k"", this shows that . Davenport showed that in 1939, by demonstrating that any sufficiently large number congruent to 1 through 14 mod 16 could be written as"
"18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, ... (The quotient can be any natural number, and the quotient = 1 if and only if ""n"" is a prime or a Carmichael number (561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, ... ), the quotient = 2 if and only if ""n"" is in the sequence: 4, 6, 15, 91, 703, 1891, 2701, 11305, 12403, 13981, 18721, ... ) The least number with ""n"" values of ""b"" are (or 0 if no such"
"1729 (number) 1729 is the natural number following 1728 and preceding 1730. It is known as the Hardy-Ramanujan number, after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. He related their conversation: The two different ways are: The quotation is sometimes expressed using the term ""positive cubes"", since allowing negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a divisor of 1729): Numbers that are the smallest number that can be expressed as the sum of two cubes in ""n"" distinct ways"
"Square-1 (puzzle) The Square-1, also known as Back to Square One and Cube 21, is a puzzle similar to the Rubik's Cube. Its distinguishing feature among the numerous Rubik's Cube variants is that it can change shape as it is twisted, due to the way it is cut, thus adding an extra level of challenge and difficulty. The Super Square One and Square Two puzzles have been recently introduced. The Super Square One has two additional layers that can be scrambled and solved independently of the rest of the puzzle, and the Square Two has extra cuts made to the"
"Least common multiple In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ""a"" and ""b"", usually denoted by LCM(""a"", ""b""), is the smallest positive integer that is divisible by both ""a"" and ""b"". Since division of integers by zero is undefined, this definition has meaning only if ""a"" and ""b"" are both different from zero. However, some authors define LCM (""a"",0) as 0 for all ""a"", which is the result of taking the LCM to be the least upper bound in the lattice of divisibility. The LCM is the ""lowest"
"Frost referred to all but the simple class as Nasik cubes. <br>The smallest normal diagonal magic cube is order 5. See Diagonal magic cube. Minimum correct summations required = 3""m"" + 6""m"" + 4 All 4m pantriagonals must sum correctly (that is 4 one-segment, 12(""m""-1) two-segment, and 4(""m""-2)(""m""-1) three-segment). There may be some simple AND/OR pandiagonal magic squares, but not enough to satisfy any other classification. <br>The smallest normal pantriagonal magic cube is order 4. See Pantriagonal magic cube. <br>Minimum correct summations required = 7""m"". All pan-""r""-agonals sum correctly for ""r"" = 1 and 3. A cube of this class"
"36 (number) 36 (thirty-six) is the natural number following 35 and preceding 37. 36 is both the square of six and a triangular number, making it a square triangular number. It is the smallest square triangular number other than one, and it is also the only triangular number other than one whose square root is also a triangular number. It is also a circular number- a square number that ends with the same integer by itself (6×6=36). It is the smallest number ""n"" with exactly eight solutions to the equation φ(""x"") = ""n"". Being the smallest number with exactly nine"
"""x"" such that where ""x"" = ""x"", ""x"" = ""x"". Within base 10 one can readily test for divisibility by 25 by seeing if the last two digits of the number match 00, 25, 50 or 75. Twenty five is: 25 (number) 25 (twenty-five) is the natural number following 24 and preceding 26. It is a square number, being 5 = 5 × 5. It is one of two two-digit numbers whose square and higher powers of the number also ends in the same last two digits, e.g. 25 = 625, the other is 76. It is the smallest square"