The **proof** **of** the **formula** is very simple. It follows straightforward from the direct calculations: . As you see, the distributive and the commutative properties of addition and multiplication operations over the real numbers are used in derivation the **formula**. The **square** **of** the **sum** **formula** is useful in a number of applications. crochet retreats near me 1. "OLS" stands for "ordinary least **squares**" while "MLE" stands for "maximum likelihood estimation.".2. The ordinary least **squares**, or OLS, can also be called the linear least.Iteratively reweighted least **squares** (IRLS) estimation is an iterative technique that solves a series of weighted least **squares** problems, where the weights are recomputed. Step 1.(Base case) Show the **formula** holds for n= 1. This is usually the easy part of an induction **proof**.Here, this is just, 1(1 + 1)(2· 1 + 1)1· 2· k2123= === 1. 66, k=1,.

In the case that k = 2 k=2 k = 2, Fermat's theorem on the **sum** of two **squares** says that an odd prime p p p is expressible as a **sum** of two **squares** if and only if p = 4 n + 1 p = 4n + 1 p = 4 n. We have \\begin{align*} r^\\top r &= Y^\\top Y - Y^\\top X\\hat \\beta + \\hat\\beta^\\top X^\\top r \\end{align*} Now \\begin{align*} X^\\top r &= X^\\top Y.

More Detail. In statistical data analysis the total **sum of squares** (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses. It is defined as being the **sum**, over all observations, of the squared differences of each observation from the overall mean. Total **Sum of Squares** is defined and given by the. We could have solved the above problem without using any loops using a **formula**. From mathematics, we know that **sum** **of** natural numbers is given by. n* (n+1)/2. For example, if n = 10, the **sum** would be (10*11)/2 = 55. Solving **sum** **of** n natural numbers in java!! #code #java #CodeEpisodes. pinata cat in the hat . Press enter for Accessibility for. The **proof** **of** the **formula** is very simple. It follows straightforward from the direct calculations: . As you see, the distributive and the commutative properties of addition and multiplication operations over the real numbers are used in derivation the **formula**. The **square** **of** the **sum** **formula** is useful in a number of applications. **Sum** **of** **squares**. The third column represents the squared deviation scores, (X-Xbar)², as it was called in Lesson 4. The **sum** **of** the squared deviations, (X-Xbar)², is also called the **sum** **of** **squares** or more simply SS. SS represents the **sum** **of** squared differences from the mean and is an extremely important term in statistics. Variance. Look at the first row of larger **squares** in the complement of the blue grid you get from colouring **squares** corresponding to multiples of blue. ... As we mentioned in the main article, there's a **formula** for the **sum** **of** the first integers: (2) Substituting this into the left part of expression (1) gives (3).

The **Sum** of **Squares** - Dynamic Geometric **Proof** This applet gives a dynamic **proof** of the **formula** for the **sum** of the **squares** of the first n natural numbers. We start with three times.

I want to make a function that gets the **sum** **of** the **squares** **of** its each digits. Although, I have seen some solutions in the internet, the one I have seen is "getting the **sum** **of** the **squares** **of** its digits" but given a list. For example, instead of starting at the integer 133, they use [1,3,3] as an input. The **sum**, S n, of the first n terms of an arithmetic series is given by: S n = ( n /2)( a 1 + a n ) On an intuitive level, the **formula** for the **sum** **of** a finite arithmetic series says that the **sum** **of** the entire series is the average of the first and last values, times the number of values being added. derivatives in a matrix (see Exercise 3.2). An alternative **proof** that b minimizes the **sum** **of** **squares** (3.6) that makes no use of ﬁrst and second order derivatives is given in Exercise 3.3. Summary of computations The least **squares** estimates can be computed as follows. Least **squares** estimation Step 1: Choice of variables.

Step 1.(Base case) Show the **formula** holds for n= 1. This is usually the easy part of an induction **proof**.Here, this is just, 1(1 + 1)(2· 1 + 1)1· 2· k2123= === 1. 66, k=1,.

Answer (1 of 2): AP: \displaystyle \sum_{k=1}^n\left(a+\left(k-1\right)d\right)^2 \displaystyle \sum_{k=1}^n\left(a^2+2ad\left(k-1\right)+d^2\left(k-1\right)^2\right. We have \\begin{align*} r^\\top r &= Y^\\top Y - Y^\\top X\\hat \\beta + \\hat\\beta^\\top X^\\top r \\end{align*} Now \\begin{align*} X^\\top r &= X^\\top Y.

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In statistical data analysis the total **sum** **of** **squares** (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses.For a set of observations ,, it is defined as the **sum** over all squared differences between the observations and their overall mean ¯.: = = (¯) For wide classes of linear models, the total **sum** **of** **squares** equals the explained **sum** **of**. The mathematician Euler formulated this **proof** in 1734 when he was 28 year old. W e will present Euler's **proof** step by step. Step 1. Using Taylor series expansion for, We have, So the Taylor series for the function is as follows: Step 2. Express as product of linear factors, Since the equation has the following roots, we have, Step 3.

The mathematician Euler formulated this **proof** in 1734 when he was 28 year old. W e will present Euler's **proof** step by step. Step 1. Using Taylor series expansion for, We have, So the Taylor series for the function is as follows: Step 2. Express as product of linear factors, Since the equation has the following roots, we have, Step 3.

3. **Sum** **of** Two **Squares** Problem 4 4. Counting Representations 9 5. Looking Ahead 11 5.1. **Sum** **of** Multiple **Squares** 11 5.2. Waring's Problem 11 6. Acknowledgments 12 References 12 1. Introduction We say that a positive integer n has a representation as a **sum** **of** two **squares** if n = a 2+ b for some nonnegative a,b ∈ Z. We deliberately include 0 as. Initialize another variable **sum** = 0 to store **sum** of prime numbers And speaking of"the **squares**"is referring to square each number Every positive prime has a corresponding negative prime: This is also false int sumOfDigits.

The calculations are based on the following results: There are four observations in each column. The overall mean is 2.1. The column means are 2.3 for column 1, 1.85 for column 2 and 2.15 for column 3.

Partitioning Total **Sum** **of** **Squares** • "The ANOVA approach is based on the partitioning of **sums** **of** **squares** and degrees of freedom associated with the response variable Y" • We start with the observed deviations of Y i around the observed mean Y¯ Yi−Y¯. **Proof** **of** the **Sum** **of** **Square** Numbers, May 27, 2012 GB High School Mathematics, High School Number Theory, In the first part of this series, we have counted the number of **squares** on a chessboard, and we have discovered that it is equal to the **sum** **of** the **squares** **of** the first 8 positive integers. The numbers , , and so on are called **square** numbers. **Sum** **of** **squares** regression (SSReg) SSReg = Σ (ŷᵢ - ȳ)² This **sums** the squared difference between the predicted value and the mean. In words, this measures how much of the **sum** **of** **squares** is explained by the regression line. Refer back to the previous plot to visualize this. **Sum** **of** **squares** total (SST) SST = Σ (yᵢ - ȳ)² SST = RSS + SSReg.

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The simplest **formula** I know that equates the **sum** **of** 2 **squares** to a product (0) ( a + b )² + ( a - b )² = 2· ( a ² + b ²) . Remark This **formula** is of independent interest in the context of **square** decompositions: regardless of primality it establishes for any n a one-to-one correspondence between the **square** decompositions of n and those of 2 n. In this video I show the **proof** for determing the **formula** for the **sum** of the **squares** of **"n"** consecutive integers, i.e. 1^2 + 2^2 + 3^2 +.... + n^2. This is a. It is time to factorise the **sum** of the two cubes by using the factoring **formula** for **sum** of two cubes . Take a = 4 x and b = 1. Now, substitute them in the sex pistols tv series limited edition space marines iaai fees public. STEP 2 Compute the total . 1) Determine the prime power representation of. We'd just stop right over there. Corollary 3. 4 2. But we cannot manually add if the number exceeds two. Relation to regularized least-**squares** • suppose A ∈ Rm×n is fat, full rank • deﬁne J1 = kAx −yk2, J2 = kxk2 • least-norm solution minimizes J2 with J1 = 0 • minimizer of weighted-**sum** objective J1 +µJ2 = kAx −yk2 +µkxk2 is xµ = ATA+µI −1 ATy • fact: xµ → xln as µ → 0, i.e., regularized solution converges to least ....

Answer: The **sum** **of** **squares** **formula** is generally referred by the **sum** **of** **squares** **of** first n natural numbers. It means 1 2 + 2 2 + 3 2 + + n 2 = Σ n 2. The **formula** **of** **sum** **of** **squares** is as follows: 1 2 + 2 2 + 3 2 + + n 2 = [n (n+1) (2n+1)] / 6. Q2: What is the **formula** for the **sum** **of** **squares** **of** even natural numbers?.

Residual **Sum** **Of** **Squares** - RSS: A residual **sum** **of** **squares** (RSS) is a statistical technique used to measure the amount of variance in a data set that is not explained by the regression model. The. Mathematical Induction **Proof** for the **Sum** **of** **Squares**, 24,851 views Jan 29, 2020 In this video I prove that the **formula** for the **sum** **of** **squares** for all positive integers n using the principle **of**.

880 RESONANCE ⎜ October 2015 GENERAL ⎜ ARTICLE Counting Your Way to the **Sum of Squares Formula** Shailesh A Shirali Keywords Combinatorial **proof**, algebraic **proof**, binomial coefficient, recur-sive relation, ordered pair. 1. Define your **formula** for consecutive integers. Once you've defined as the largest integer you're adding, plug the number into the **formula** to **sum** consecutive integers: **sum** = ∗ ( +1)/2. [4] For example, if you're summing the first 100 integers, plug 100 into. n {\displaystyle n} to get 100∗ (100+1)/2. Relation to regularized least-**squares** • suppose A ∈ Rm×n is fat, full rank • deﬁne J1 = kAx −yk2, J2 = kxk2 • least-norm solution minimizes J2 with J1 = 0 • minimizer of weighted-**sum** objective J1 +µJ2 = kAx −yk2 +µkxk2 is xµ = ATA+µI −1 ATy • fact: xµ → xln as µ → 0, i.e., regularized solution converges to least .... .

Answer (1 of 4): How much is the following **sum**? 1 + 2 + 3 + 4 + 5 + + 100 = _____ To answer the question, the first thing that can happen is we add up sequence. Recently I have seen several articles on arxiv that refer to a **proof** system called **sum-of-squares**. Can someone explain what is a **sum-of-squares** **proof** and why such **proofs** are important/interesting? How are they related to other algebraic **proof** systems?.

The two-way ANOVA is probably the most popular layout in the Design of Experiments. To begin with, let us define a factorial experiment : An experiment that utilizes every combination of factor levels as treatments is called a factorial experiment. Model for the two-way factorial experiment. In a factorial experiment with factor at levels and. Residual **Sum** **of** **Squares** Calculator. This calculator finds the residual **sum** **of** **squares** **of** a regression equation based on values for a predictor variable and a response variable. Simply enter a list of values for a predictor variable and a response variable in the boxes below, then click the "Calculate" button:.

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Initialize another variable **sum** = 0 to store **sum** of prime numbers And speaking of"the **squares**"is referring to square each number Every positive prime has a corresponding negative prime: This is also false int sumOfDigits. As predicted by Fermat's theorem on the **sum** **of** two **squares**, each can be expressed as a **sum** **of** two **squares**: 5=12+225 = 1^2 + 2^25=12+22, 17=12+4217 = 1^2 + 4^217=12+42, and 41=42+5241 = 4^2 + 5^241=42+52. On the other hand, odd primes 777, 191919, and 313131are all congruent to 3 mod 43 \bmod 43mod4and cannot be expressed as a **sum** **of** two **squares**.

**Proof** **of** the **Sum** **of** **Square** Numbers, May 27, 2012 GB High School Mathematics, High School Number Theory, In the first part of this series, we have counted the number of **squares** on a chessboard, and we have discovered that it is equal to the **sum** **of** the **squares** **of** the first 8 positive integers. The numbers , , and so on are called **square** numbers. If the **sum** **of squares** were not normalized, its value would always be larger for the sample of 100 people than for the sample of 20 people. To scale the **sum** **of squares**, we divide it by the degrees of freedom, i.e., calculate the **sum** **of squares** per degree of freedom, or variance. Standard deviation, in turn, is the square root of the variance.. However, we have an integral rule, which helps us to evaluate the integration of the reciprocal of **sum** **of** one and **square** **of** a variable. Hence, it is a good idea to convert the reciprocal of **sum** **of** **squares** into **sum** **of** one and **square** **of** a term. = ∫ 1 1 × ( x 2 + a 2) d x. = ∫ 1 a 2 a 2 × ( x 2 + a 2) d x. = ∫ 1 a 2 × ( x 2 + a 2 a 2) d x.

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In statistical data analysis the total **sum** **of** **squares** (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses.For a set of observations ,, it is defined as the **sum** over all squared differences between the observations and their overall mean ¯.: = = (¯) For wide classes of linear models, the total **sum** **of** **squares** equals the explained **sum** **of**. Even though it appears that the total is always **square**, the **sum** **of** the first n cubes, 13+23+ + n 3 = (n (n +1)/2)2, which is the **square** **of** the nth triangular number, is surprising. Now, 13+23+ +103= (10111/2)2=552 = 3025, for example. Apr 21, 2021 · **Proof** of \((a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca\) To prove the above **formula**, take a square with side a+b+c units, as shown in the figure. The bigger square is separated into nine quadrilaterals i.e. a combination of rectangles+**squares**.. **Sums** **of** independent random variables. by Marco Taboga, PhD. This lecture discusses how to derive the distribution of the **sum** **of** two independent random variables.We explain first how to derive the distribution function of the **sum** and then how to derive its probability mass function (if the summands are discrete) or its probability density function (if the summands are continuous). Step 1.(Base case) Show the **formula** holds for n= 1. This is usually the easy part of an induction **proof**.Here, this is just, 1(1 + 1)(2· 1 + 1)1· 2· k2123= === 1. 66, k=1,. In statistical data analysis the total **sum** **of** **squares** (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses.For a set of observations ,, it is defined as the **sum** over all squared differences between the observations and their overall mean ¯.: = = (¯) For wide classes of linear models, the total **sum** **of** **squares** equals the explained **sum** **of**. Thus, the divisor **sum** **of** f evaluated at a positive integer n takes the positive divisors of n, plugs them into f, and adds up the results. A similar convention will hold for products. Notice that the divisor **sum** is a function which takes an arithmetic function as input and produces an arithmetic function as output . Example.

The **sum** of **squares** is one of the most important outputs in regression analysis. The general rule is that a smaller **sum** of **squares** indicates a better model, as there is less.

**sum** **of** **squares** **formula** proofsum of **squares** **formula** **proof**. May 12, 2022 best day trips from london in winter.

A natural numberncould be written as a **sum** **of** the **squares** **of** two integers if and only if every prime factorpofnwhich is of the form 4k+3 enters the canonical decomposition ofnto an even degree. Examples: 306 = 2⁄32⁄17 is **sum** **of** two **squares**, while 102 = 2⁄3⁄17 is not. **Proof**. Letnbe a number with factorization of the kind described in the the- orem.

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11 **Sum** of **Squares** S. Lall, Stanford 2011.04.18.01 convexity the sets of PSD and SOS polynomials are a convex cones; i.e., f,g PSD =⇒ λf +µg is PSD for all λ,µ ≥ 0 let Pn,d be the. Products of **Sums** **of** Two **Squares** Here's a nice theorem due to Fibonacci, in 1202. Theorem. If integers N and M can each be written as the **sum** **of** two **squares**, so can their product! Example: since 2=1 2 +1 2 and 34=3 2 +5 2, their product 68 should be expressible as the **sum** **of** two **squares**. In fact, 68=8 2 +2 2.

Sep 08, 2020 · The **formula** Y = a + bX. The **formula**, for those unfamiliar with it, probably looks underwhelming – even more so given the fact that we already have the values for Y and X in our example. Having said that, and now that we're not scared by the **formula**, we just need to figure out the a and b values. To give some context as to what they mean:.

An Interesting Equality for **Sum** **of** Reciprocals of the **Squares** P ... Overview Some History about the **Sum** Review: Maclaurin Series Euler's "**Proof**" ... previous **formula**. Choe's **Proof** (1) Corollary For |t| < π/2,. We can find the **sum** **of** **squares** **of** the first n natural numbers using the **formula**, **SUM** = 1 2 + 2 2 + 3 2 + ... + n 2 = [n (n+1) (2n+1)] / 6. We can prove this **formula** using the principle of mathematical induction. Let's go through the **formulas** **of** finding the **sum** **of** **squares** **of** even and odd natural numbers in the next section. I Integers of the form 3n + 1 as **sums** **of** three **squares** studied by Diophantus (200-300 AD). I Fermat, Euler, Lagrange, Legendre, and Dirichlet all studied the problem. I First **proof** **of** the three-**square** theorem published by Legendre in 1798. I Clearer **proof** presented by Dirichlet in 1850 based on the theory of binary and ternary quadratic forms. **Pythagoras theorem** states that “In a right-angled triangle, the square of the hypotenuse side is equal to the **sum** **of squares** of the other two sides“. The sides of this triangle have been named Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°. The sides of a right triangle (say ....

The **sums** **of** **squares** for explanatory variable A is harder to see in the **formula** , but the same reasoning can be used to understand the denominator for forming the Mean **Square** for variable A or MS A: there are J means that vary around the grand mean so MS A = SS A /(J-1). In summary, the two mean **squares** are simply:. Question 1. Find the **sum** **of** the **squares** **of** the first 20 natural numbers. Solution. We know that the **formula** for the **sum** **of** the **squares** **of** the first n natural numbers, the **formula** is, Substituting for the above equation, we get, Upon simplification, we get, Question 2. Find the **sum** **of** the **squares** **of** the first 18 odd numbers.

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We can find the **sum** **of** **squares** **of** the first n natural numbers using the **formula**, **SUM** = 1 2 + 2 2 + 3 2 + ... + n 2 = [n (n+1) (2n+1)] / 6. We can prove this **formula** using the principle of mathematical induction. Let's go through the **formulas** **of** finding the **sum** **of** **squares** **of** even and odd natural numbers in the next section.

An Interesting Equality for **Sum** **of** Reciprocals of the **Squares** P ... Overview Some History about the **Sum** Review: Maclaurin Series Euler's "**Proof**" ... previous **formula**. Choe's **Proof** (1) Corollary For |t| < π/2,. The **sum** **of** n consecutive cubes is equal to the **square** **of** the nth triangle. 1 3 + 2 3 + 3 3 + . . . + n 3 = (1 + 2 + 3 + . . . + n) 2. To see that, we will begin here: The difference between the **squares** **of** two consecutive triangular numbers is a cube. Triangles: 1 3 6 10 15 21 28.

**sum** **of** **squares** and sem, ideﬁnite programm, supposef, letz, is , 2R[x , 1, vector of , if and . . . , all , xn], of degree2d, monomials of degree , be , SOS , this , the less , only if there , existsQsuch that, is an SDP , number of in standard , Qº0, f=z, primal , components ofz, Qz, form, n+d, is, d, comparing terms gives aﬃne constraints on ,. This is the Total **Sum** **of** **Squares** Error! Therefore, we can see that the deviance is a generalised **formula** for the (Scaled) **Sum** **of** **Squares** Error for Linear Regression. We can carry out a similar derivation to other probability distributions such as the Poisson, Gamma and Binomia l distribution using their PDFs to calculate their deviance.

**Sum** **of** **squares** (SS) is a statistical tool that is used to identify the dispersion of data as well as how well the data can fit the model in regression analysis. The **sum** **of** **squares** got its name because it is calculated by finding the **sum** **of** the squared differences. This image is only for illustrative purposes. Euler's Pi for the **Sum** **of** Inverse **Squares** **Proof**, 2 minute read, Published:May 29, 2020, Given an infinite series of inverse **squares** **of** the natural numbers, what is the **sum**? X = 1/(1^2) + 1/(2^2) + 1/(3^2) + 1/(4^2) + ... The above is the Basel problem that asks for the precise **sum** **of** the inverse **squares** **of** the natural numbers. Finding the **sum of squares** in Microsoft Excel can be a repetitive task. The most obvious **formula** requires a lot of data entry, though there’s a lesser-known option that gets you to the same place. Join 425,000 subscribers and get a.

We can try another approach, and look for the **sum** **of** the **squares** **of** the first n natural numbers, hoping that this **sum** will vanish. Second Try With Summation Starting again, we note that the **sum** **of** the **squares** **of** the first n natural numbers is the **sum** **of** the first (n+1), less (n+1) 2. Expanding the (k+1)th term:. In the population, the **formula** is. SSY = ∑(Y − μY)2. where SSY is the **sum** **of** **squares** Y, Y is an individual value of Y, and μy is the mean of Y. A simple example is given in Table 14.3.1. The mean of Y is 2.06 and SSY is the **sum** **of** the values in the third column and is equal to 4.597. Table 14.3.1: Example of SSY. Y.

Initialize another variable **sum** = 0 to store **sum** of prime numbers And speaking of"the **squares**"is referring to square each number Every positive prime has a corresponding negative prime: This is also false int sumOfDigits. that minimizes the **sum** **of** squared residuals, we need to take the derivative of Eq. 4 with respect to. ﬂ^. This gives us the following equation: @e. 0. e @ﬂ ^ = ¡ 2. X. 0. y +2. X. 0. Xﬂ ^ = 0 (5) To check this is a minimum, we would take the derivative of this with respect to. ﬂ^ again { this gives us 2. X. 0. X.

Products of **Sums** **of** Two **Squares** Here's a nice theorem due to Fibonacci, in 1202. Theorem. If integers N and M can each be written as the **sum** **of** two **squares**, so can their product! Example: since 2=1 2 +1 2 and 34=3 2 +5 2, their product 68 should be expressible as the **sum** **of** two **squares**. In fact, 68=8 2 +2 2.

The variance is the square of the standard deviation, the second central moment of a distribution, and the Example of samples from two populations with the same.

So I want figure out an algebraic **proof** **of** the **sum** **of** **squares** **formula**. All perfect **squares** are the summation of n successive odd terms (4 2 = 1+3+5+7). I'm trying to use this fact to derive the **formula**, any ideas?.

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Step 1.(Base case) Show the **formula** holds for n= 1. This is usually the easy part of an induction **proof**.Here, this is just, 1(1 + 1)(2· 1 + 1)1· 2· k2123= === 1. 66, k=1,. However, we have an integral rule, which helps us to evaluate the integration of the reciprocal of **sum** of one and square of a variable. Hence, it is a good idea to convert the reciprocal of **sum**. Figure 8: The pyramid of **squares**. The **Formula** for the **Sum** **of** **Squares**. The **formula** for the **sum** **of** **squares** may not have been new to Archimedes, and there is evidence that it might have been discovered about the same time in India. We do know that it was rediscovered many times. The earliest **proofs**, including Archimedes's **proof**, are all geometric.

Also Read: **Sum** **of** **Squares** **Formula**, **Proof** For **Sum** **of** Cubes **Formula**, [Click Here for Sample Questions] The **proof** or the verification of the **formula** **of** the **sum** **of** cubes {a 3 + b 3 = (a + b) (a 2 - ab + b 2 } is given by proving that LHS = RHS here. LHS = a 3 + b 3, After the solution of RHS term, we find, = (a + b) (a 2 - ab + b 2). S (200) - S (100) + 100² = 200 (201) (401)/6 + 100 (101) (201) + 100² = 2686700 - 338350 + 10000 = 2358350 Fun Facts About **Square** Pyramid Numbers 1. The only two **square** pyramidal numbers that are also **squares** are 1 and 4900, which is the **sum** **of** the first 24 **squares**. 2. **Proof** **of** the **Sum** **of** **Square** Numbers, May 27, 2012 GB High School Mathematics, High School Number Theory, In the first part of this series, we have counted the number of **squares** on a chessboard, and we have discovered that it is equal to the **sum** **of** the **squares** **of** the first 8 positive integers. The numbers , , and so on are called **square** numbers.