Square Of Numbers From 1 To 50 Pdf
(6)As a part of the study of, it is known that every positive integer is a sum of no more than 4 positive squares(; ), that every 'sufficiently large' integer is asum of no more than 4 positive squares ( ), and thatevery integer is a sum of at most 3 signed squares ( ). Actually,the basis set for representing positive integers with positive squares is,so 49 need never be used. Furthermore, since an infinite number of require four squaresto represent them, the least such that everybeyond a certain point requiressquares is given by.The number of representation of a number by squares, distinguishingsigns and order, is denoted and calledthe. The minimumnumber of squares needed to represent the numbers 1, 2, 3. Are 1, 2, 3, 1, 2,3, 4, 2, 1, 2.
(OEIS ), and thenumber of distinct ways to represent the numbers 1, 2, 3. In terms of squaresare 1, 1, 1, 2, 2, 2, 2, 3, 4, 4. (OEIS ).A brute-force algorithm for enumerating the square partitions of is repeated applicationof the.
Square Of Numbers From 1 To 50 Pdf Converter
However, this approachrapidly becomes impractical since the number of representations grows extremely rapidlywith, as shown in the following table.squarepartitionsThe th nonsquare number is given. (13)has the same last two digits as (with theone additional possibility that in which casethe last two digits are 00). The following table (with the addition of 00) thereforeexhausts all possible last two digits.364964811214469962556892461241842976257629844136124895625966944214816449362516090401The only 22 possibilities are therefore 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, and 96, which can be summarized succinctlyas 00, 25, and, where stands for an and for an. Additionally, a (but not )condition for a number to be square is that its be 1, 4, 7, or 9. The digital roots of the first few squares are 1, 4, 9,7, 7, 9, 4, 1, 9, 1, 4, 9, 7.
(OEIS ),while the list of number having digital roots 1, 4, 7, or 9 is 1, 4, 7, 9, 10, 13,16, 18, 19, 22, 25. (OEIS ).This property of square numbers was referred to in a 'puzzler' feature of a March 2008 broadcast of the NPR radio show 'Car Talk.'
In this Puzzler,a son tells his father that his computer and math teacher assigned the class a problemto determine if a number is a perfect square. Each student is assigned a particularnumber, and the students are supposed to write a software program to determine theanswer. The son's assigned number was.While the father thinks this is a hard problem, a bystander listening to the conversationstates that the teacher gave the son an easy number and the bystander can give theanswer immediately. The question is what does this third person know? The answeris that the number ends in the digit '2,' which is not one of the possiblelast digits for a square number.The following table gives the possible residues mod for square numbersfor to 20. The quantity gives the numberof distinct residues for a given.220,1320,1420,1530,1, 4640, 1, 3, 4740, 1, 2, 4830, 1, 4940, 1, 4, 71060, 1, 4, 5, 6, 91160, 1, 3, 4, 5, 91240, 1, 4, 91370,1, 3, 4, 9, 10, 121480, 1, 2, 4, 7, 8, 9, 111560, 1, 4, 6, 9, 101640, 1, 4, 91790,1, 2, 4, 8, 9, 13, 15, 161880, 1, 4, 7, 9, 10, 13, 1619100, 1, 4, 5, 6, 7,9, 11, 16, 172060, 1, 4, 5, 9, 16In general, the squares are congruent to 1 (mod 8) (Conway and Guy 1996).
Square Of Numbers From 1 To 50 Pdf File
Stangl (1996) gives an explicit formula by which the numberof squares in (i.e., mod ) can be calculated. Then is the given. (33)(Robertson 1996).states that 8 and 9 ( and ) are the onlyconsecutive (excluding 0 and 1), i.e., the onlysolution to.This has not yet been proved or refuted,although R. Tijdeman has proved that there can be only a finite number of exceptionsshould the not hold. It is also known that8 and 9 are the only consecutive and squarenumbers (in either order).The numbers that are not the difference of two squares are 2, 6, 10, 14, 18.(OEIS; Wells 1986, p. 76).A square number can be the concatenation of two squares, as in the case and giving. The firstfew numbers that are neither square nor the sum of a square and aare 10, 34, 58, 85, 91, 130, 214. (OEIS ).It is conjectured that, other than, and, there are only anumber of squares having exactly two distinct (Guy 1994, p. 262).
The first few such are 4, 5, 6, 7,8, 9, 11, 12, 15, 21. (OEIS ), correspondingto of 16, 25, 36, 49, 64, 81, 121.(OEIS ).The following table gives the first few numbers which, when squared, give numbers composed of only certain digits. The values of such that contains exactly two different digits are givenby 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20. (OEIS ),whose squares are 16, 25 36, 49, 64.
(OEIS ).digitsSloane,1, 2, 31, 11, 111, 36361, 363639.1,121, 12321,.1, 4, 61,2, 4, 8, 12, 21, 38, 108.1, 4, 16, 64, 144,441, 1444.1, 4, 91, 2, 3, 7, 12, 21,38, 107.1, 4, 9, 49, 144, 441, 1444, 11449.2, 4, 82, 22, 168, 478, 2878, 210912978.4, 484, 28224, 228484, 8282884.4,5, 62,8, 216, 238, 258, 738, 6742.4,64, 46656, 56644, 66564.For three digits, an extreme example containing only the digits 7, 8, and 9 is. Wolfram Web ResourcesThe #1 tool for creating Demonstrations and anything technical.Explore anything with the first computational knowledge engine.Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.Join the initiative for modernizing math education.Solve integrals with Wolfram Alpha.Walk through homework problems step-by-step from beginning to end.
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MeaningInformally: When you multiply a whole number times itself, the resulting product is called a square number, or a perfect square or simply 'a square.' So 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on, are all square numbers.A more formal definition: A square number is a number of the form (or n2), where n is any whole number. Mathematical background Objects arranged in a square arrayThe name 'square number' comes from the fact that these particular numbers of objects can be arranged to fill a perfect square.Children can experiment with pennies (or square tiles) to see what numbers of them can be arranged in a perfectly square array.Four pennies can: Nine pennies can:And sixteen pennies can, too:But seven pennies or twelve pennies cannot be arranged that way. Numbers (of objects) that can be arranged into a square array are called 'square numbers.' Not a square number: Square arrays must be 'full' if we are to count the number as a 'square number.' Here are 12 pennies arranged in a square, but not a 'full' square array.The number 12 is not a square number.Children may enjoy exploring what numbers of pennies can be arranged into an open square like this.
They are not called 'square numbers' but do follow an interesting pattern.Squares made of square tiles are also fun to make. Again, the number of square tiles that fit into a square array is a 'square number.' In the multiplication tableSquare numbers appear along the diagonal of a standard multiplication table. Connections with triangular numbersIf you count the green triangles in each of these designs, the sequence of numbers you see is: 1, 3, 6, 10, 15, 21., a sequence called (appropriately enough) the triangular numbers. If you count the white triangles that are in the 'spaces' between the green ones, the sequence of numbers starts with 0 (because the first design has no gaps) and then continues: 1, 3, 6, 10, 15., again triangular numbers!Remarkably, if you count all the tiny triangles in each design - both the green and the white - the numbers are square numbers!This connection between square numbers and triangular numbers can be seen another way, too.Build a stair-step arrangement of Cuisenaire rods, say W, R, G. Then build the very next stair-step: W, R, G, P.Each is 'triangular' (if we ignore the stepwise edge). Put the two consecutive triangles together, and they make a square:.
This square is the same size as 16 white rods arranged in a square. The number 16 is a square number, '4 squared,' the square of the length of the longest rod (as measured with white rods).Here's another example:. When placed together, these make a square whose area is 64, again the square of the length (in white rods) of the longest rod. (The brown rod is 8 white rods long, and 64 is 8 times 8, or '8 squared.' ) Stair steps from square numbers. Stair steps that go up and then back down again, like this, also contain a square number of tiles. When the tiles are checkerboarded, as they are here, an addition sentence that describes the number of red tiles (10), the number of black tiles (6), and the total number of tiles (16) shows, again, the connection between triangular numbers and square numbers: 10 + 6 = 16.Inviting children in grade 2 (or even 1) to build stair-step patterns and write number sentences that describe these patterns is a nice way to give them practice with descriptive number sentences and also becoming 'friends' with square numbers.