Linear Relations between Numbers of Terms and First Terms of Sums of Consecutive Squared Integers Equal to Squared Integers

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Linear Relations between Numbers of Terms and First Terms of Sums of Consecutive Squared Integers Equal to Squared Integers

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Vladimir Pletser^*

This version is not peer-reviewed

Submitted:

29 December 2023

Posted:

29 December 2023

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Abstract

Sums of M consecutive squared integers (a+i)^2 equaling squared integers (for a<=1, 0<= i<= M-1) yield remarkable regular linear features when plotting values of M in function of a. These features correspond to groupings of pairs of a values for successive same values of M around straight lines of equation mu*M = 2a and are characterized in this paper for rational values of mu.

Keywords:

Subject: Computer Science and Mathematics - Algebra and Number Theory

1. Introduction

The study of integer squares equal to sums of consecutive squared integers can be dated back to 1873 when Lucas stated [11] that

(1^{2} + . . . + n^{2})

is an integer square only for

n = 1

and 24. Lucas proposed later in 1875 [12] the well known cannonball problem, which was proven by several authors [2,10,13,14,19,20,33].

Instead of starting at 1, finding all values of a for which the sum of M consecutive integer squares starting from

a^{2} \geq 1

is itself an integer square

s^{2}

is a more general problem that was addressed by several authors (see e.g. [1,3,8,24]). More recently, this author showed [26] that there are no integer solutions if

M \equiv 3, 5, 6, 7, 8

10 (m o d 12)

and that there are integer solutions for non squared integer M congruent to

0, 9, 24

33 (m o d 72)

, or to

1, 2

16 (m o d 24)

, or to

11 (m o d 12)

, and for squared integer M congruent to

1 (m o d 24)

In this paper, we investigate and characterize the properties of groupings of pairs of a values for a same value of M that are found around inclined straight lines of equation

μ M \approx 2 a

in the

(a, M)

plot for rational values of

μ

2. Linear features in the $(a, M)$ plot

For

M > 1, a, i, s \in Z^{*}

, the sum of M consecutive squared integers

{(a + i)}^{2}

equaling a squared integer

s^{2}

can be written [28] as

\sum_{i = 0}^{M - 1} {(a + i)}^{2} = M [{(a + \frac{M - 1}{2})}^{2} + \frac{M^{2} - 1}{12}]

(1)

For

1 \leq a \leq 10^{5}

and

2 \leq M \leq 10^{5}

, there are only 4078 couples of values of a and M among the approximately

10^{10}

possibilities such that (1) holds. Figure 1 shows the distribution of these 4078 couples in a

(a, M)

plot where several groupings of interest are seen.

The most visible is the grouping around a straight line of equation

M = 2 a + c

where c is a constant, corresponding to a double infinite family of a values that starts with the identity and the Pythagorean relations

\begin{matrix} 0^{2} + 1^{2} & = & 1^{2} \end{matrix}

(2)

\begin{matrix} 3^{2} + 4^{2} & = & 5^{2} \end{matrix}

(3)

for a same value of

M = 2

and respectively for

a = 0

and 3. This double infinite family has the characteristics that couples of a values correspond to a same value of M. There are other similar groupings and double infinite families around straight lines of general equation

μ M = 2 a + c_{μ}

for certain rational values of

μ > 0

and where

c_{μ}

is a constant different for each value of

μ

. Only groupings around inclined lines are considered as the limit cases of

μ = 0

and

μ \to \infty

corresponding to groupings around respectively vertical and horizontal lines are not treated here. The horizontal case for which one or several solutions in a exist for each values of M was investigated in [26,27,28].

Definition 1.

For

1 \leq j \leq 2, M_{μ, k} > 1, a_{j, μ, k} \in Z^{*}

, for a given value of

μ \in Q^{+}

, two values of

a_{j, μ, k}

are called a pair

(a_{1, μ, k}, a_{2, μ, k})

if for a same value of

M_{μ, k}

and

\forall k \in Z

\begin{matrix} a_{1, μ, k} + a_{2, μ, k} & = & μ M_{μ, k} + 1 \end{matrix}

(4)

\begin{matrix} a_{2, μ, k} - a_{1, μ, k} & = & f_{μ, k} \end{matrix}

(5)

hold, where

f_{μ, k} = f_{μ} (k)

is a linear integer function of k for each value of μ, yielding

μ M_{μ, k} = 2 a_{j, μ, k} \pm f_{μ, k} - 1

(6)

where the upper or lower sign is taken for

j = 1

or 2.

The two families of

a_{1, μ, k}

and

a_{2, μ, k}

are characterized for each values of

μ

around the straight line of equation

μ M = 2 a + c_{μ}

in the following theorem. However, relations (4) to (6) hold only for certain values, called allowed values, of

M_{μ, k}

and of

μ

that are determined further.

Theorem 1.

For

1 \leq j \leq 2, i, η, δ, M_{μ, k} > 1, a_{j, μ, k} \in Z^{*}

k, s_{j, μ, k} \in Z

, for allowed values of

μ \in Q^{+}

, let

μ = (η / δ)

be an irreducible fraction; if

(a_{1, μ, k}, a_{2, μ, k})

is a pair of

a_{j, μ, k}

values for a same value of

M_{μ, k}

and if

M_{μ, k} = \frac{(3 f_{μ, k}^{2} - 1)}{3 {(μ + 1)}^{2} + 1} = \frac{δ^{2} (3 f_{μ, k}^{2} - 1)}{3 {(η + δ)}^{2} + δ^{2}}

(7)

holds

\forall k \in Z

, then the sums of squares of

M_{μ, k}

consecutive integers

(a_{j, μ, k} + i)

for

i = 0

M_{μ, k} - 1

are always equal to squared integers

s_{j, μ, k}^{2}

, with

\begin{matrix} a_{j, μ, k} & = & \frac{1}{2 δ} (η M_{μ, k} + δ \mp \sqrt{\frac{M_{μ, k} (3 {(η + δ)}^{2} + δ^{2}) + δ^{2}}{3}}) \end{matrix}

(8)

\begin{matrix} s_{j, μ, k} & = & \frac{M_{μ, k}}{2 δ} (\sqrt{\frac{M_{μ, k} (3 {(η + δ)}^{2} + δ^{2}) + δ^{2}}{3}} \mp (η + δ)) \end{matrix}

(9)

where the upper (respectively lower) sign is taken for

j = 1

(resp. 2).

Proof.

Let

1 \leq j \leq 2, i, η, δ, M_{μ, k} > 1, a_{j, μ, k} \in Z^{*}

k, s_{j, μ, k} \in Z

μ = (η / δ) \in Q^{+}

forming an irreducible fraction, i.e.

gcd (η, δ) = 1

. Let further

f_{μ, k}

be a yet unknown integer function of k for each value of

μ

. Replacing in the second equality of (1) M by

M_{μ, k}

and a by

a_{j, μ, k}

from (6) yields successively

\begin{matrix} \sum_{i = 0}^{M_{μ, k} - 1} {(a_{j, μ, k} + i)}^{2} & = & \frac{M_{μ, k}}{4} [{(2 a_{j, μ, k} + M_{μ, k} - 1)}^{2} + \frac{M_{μ, k}^{2} - 1}{3}] \\ = & \frac{M_{μ, k}}{4} [{((μ + 1) M_{μ, k} \mp f_{μ, k})}^{2} + \frac{M_{μ, k}^{2} - 1}{3}] \\ = & \frac{M_{μ, k}^{2}}{4} [\frac{(3 {(η + δ)}^{2} + δ^{2}) M_{μ, k}}{3 δ^{2}} + \frac{3 f_{μ, k}^{2} - 1}{3 M_{μ, k}} \\ \mp 2 (\frac{η + δ}{δ}) f_{μ, k}] \end{matrix}

(10)

where the upper (respectively lower) sign in (10) is taken for

j = 1

(resp. 2). For the expression between brackets in (10) to be a square, replace in (10)

f_{μ, k}

f_{μ, k} = \sqrt{\frac{M_{μ, k} (3 {(η + δ)}^{2} + δ^{2}) + δ^{2}}{3 δ^{2}}}

(11)

from (7), yielding immediately (9). Replacing

f_{μ, k}

(11) in (6) yields then (8). □

In addition, from Theorem 2, the following relations hold

\forall k \in Z

\begin{matrix} a_{2, μ, k} - a_{1, μ, k} & = & \sqrt{\frac{M_{μ, k} (3 {(η + δ)}^{2} + δ^{2}) + δ^{2}}{3 δ^{2}}} \end{matrix}

(12)

\begin{matrix} s_{2, μ, k} + s_{1, μ, k} & = & M_{μ, k} \sqrt{\frac{M_{μ, k} (3 {(η + δ)}^{2} + δ^{2}) + δ^{2}}{3 δ^{2}}} \end{matrix}

(13)

\begin{matrix} = & M_{μ, k} (a_{2, μ, k} - a_{1, μ, k}) \end{matrix}

(14)

\begin{matrix} s_{2, μ, k} - s_{1, μ, k} & = & M_{μ, k} (\frac{η + δ}{δ}) \end{matrix}

(15)

\begin{matrix} = & a_{2, μ, k} + a_{1, μ, k} + M - 1 \end{matrix}

(16)

3. Parametric expressions of $f_{μ, k}$ , $M_{μ, k}$ , $a_{j, μ, k}$ , $s_{j, μ, k}$

Above results hold only for certain allowed values of

M_{μ, k}

and of

μ \in Q^{+}

, that can be determined as follows. Relation (7) reads also

{(δ f_{μ, k})}^{2} - M_{μ, k} {(η + δ)}^{2} = δ^{2} (\frac{M_{μ, k} + 1}{3})

(17)

It was shown [30] that for (17) to hold:

δ \equiv 0 (m o d 6)

, and

η \equiv 1

5 (m o d 6)

M_{μ, k} \equiv 0

24 (m o d 72)

, and

M_{μ, k} (m o d (δ^{2} / 3)) \equiv 0

;

δ \equiv 1

5 (m o d 6)

, and

η \equiv 1, 3

5 (m o d 6)

and either

f_{μ, k} \equiv 1 (m o d 2)

and

M_{μ, k} \equiv 2 (m o d 24)

, or

f_{μ, k} \equiv 0 (m o d 2)

and

M_{μ, k} \equiv 11 (m o d 12)

, and

M_{μ, k} (m o d δ^{2}) \equiv 0

Parametric expressions of

f_{μ, k}

M_{μ, k}

a_{j, μ, k}

and

s_{j, μ, k}

in function of

k \in Z

μ = (η / δ)

and initial values are found as follows.

Theorem 2.

For

1 \leq j \leq 2, η, δ, M_{μ, k} > 1 \in Z^{+}

a_{j, μ, k} \in Z^{*}

k, s_{j, μ, k} \in Z

μ, ν \in Q^{+}

, for allowed values of

μ = (η / δ)

and for pairs

(a_{1, μ, k}, a_{2, μ, k})

f_{μ, k}

is a linear function of k,

M_{μ, k}

and

a_{j, μ, k}

are quadratic functions of k, and

s_{j, μ, k}

is a cubic function of k, as follows

\begin{matrix} f_{μ, k} & = & (3 {(η + δ)}^{2} + δ^{2}) ν k + f_{μ, 0} \end{matrix}

(18)

\begin{matrix} M_{μ, k} & = & 3 δ^{2} (3 {(η + δ)}^{2} + δ^{2}) ν^{2} k^{2} + 6 δ^{2} f_{μ, 0} ν k + M_{μ, 0} \\ a_{j, μ, k} & = & \frac{1}{2} [3 η δ (3 {(η + δ)}^{2} + δ^{2}) ν^{2} k^{2} + (6 η δ f_{μ, 0} \mp (3 {(η + δ)}^{2} + δ^{2})) ν k] \end{matrix}

(19)

\begin{matrix} + a_{j, μ, 0} \\ s_{j, μ, k} & = & \frac{1}{2} [3 δ^{2} {(3 {(η + δ)}^{2} + δ^{2})}^{2} ν^{3} k^{3} \\ + 3 δ (3 {(η + δ)}^{2} + δ^{2}) (3 δ f_{μ, 0} \mp (η + δ)) ν^{2} k^{2} \end{matrix}

(20)

\begin{matrix} + ({(3 δ f_{μ, 0} \mp (η + δ))}^{2} - ({(η + δ)}^{2} + δ^{2})) ν k] + s_{j, μ, 0} \end{matrix}

(21)

where

ν = 1

for

δ \equiv 1

5 (m o d 6)

and

ν = (2 / 3)

for

δ \equiv 0 (m o d 6)

and where the upper (respectively lower) sign is taken for

j = 1

(resp. 2).

Proof.

For

1 \leq j \leq 2, η, δ, x, M_{μ, k} > 1 \in Z^{+}

a_{j, μ, k} \in Z^{*}

k, s_{j, μ, k} \in Z

μ, ν \in Q^{+}

, for allowed values of

μ = (η / δ)

and for pairs

(a_{1, μ, k}, a_{2, μ, k})

, let

f_{μ, k}

(5) be a linear function of k,

f_{μ, k} = x k + f_{μ, 0}

where

f_{μ, 0} = (a_{2, μ, 0} - a_{1, μ, 0})

is the initial value for

k = 0

of the difference (5) and x an integer function to be defined for some parameters. Then (7) yields

\begin{matrix} M_{μ, k} & = & \frac{δ^{2} (3 {(x k + f_{μ, 0})}^{2} - 1)}{3 {(η + δ)}^{2} + δ^{2}} \\ = & \frac{3 δ^{2} x k (x k + 2 f_{μ, 0})}{3 {(η + δ)}^{2} + δ^{2}} + \frac{δ^{2} (3 f_{μ, 0}^{2} - 1)}{3 {(η + δ)}^{2} + δ^{2}} \end{matrix}

(22)

The second term on the right of (22) is

M_{μ, 0}

by (7).

(i) If

δ \equiv 1

5 (m o d 6)

, as

M_{μ, k} \in Z^{+}

(3 {(η + δ)}^{2} + δ^{2})

must divide x in the first term of (22), yielding then (18) and (19) with

ν = 1

(ii) If

δ \equiv 0 (m o d 6)

, simplifying the first term by 3, (22) reads

M_{μ, k} = \frac{δ^{2} x k (x k + 2 f_{μ, 0})}{{(η + δ)}^{2} + (δ^{2} / 3)} + M_{μ, 0}

(23)

M_{μ, k} \in Z^{+}

({(η + δ)}^{2} + (δ^{2} / 3))

is a factor of x. However, as

η \equiv 1

5 (m o d 6)

for

δ \equiv 0 (m o d 6)

({(η + δ)}^{2} + (δ^{2} / 3)) \equiv 1 (m o d 2)

and as

f_{μ, k} (m o d 2) \equiv f_{μ, 0} (m o d 2)

\forall k \in Z

, x must be replaced by

2 ({(η + δ)}^{2} + (δ^{2} / 3))

, yielding then (18) and (19) with

ν = (2 / 3)

(iii) Further, replacing

f_{μ, k}

and

M_{μ, k}

by (18) and (19) in

a_{j, μ, k}

from (6) and in

s_{j, μ, k}

(9) with (11), yield directly (20) and (21) with the upper (or lower) sign for

j = 1

(or 2). □

4. Finding allowed values of $μ = (η / δ)$ and $M_{μ, k}$

Finding the allowed values of

μ = (η / δ)

M_{μ, 0}

and

f_{μ, 0}

requires solving the generalized Pell equation (17) for

k = 0

in variables

(δ f_{μ, 0})

and

(η + δ)

In general, for

X, Y, D, N, x_{f}, y_{f}, n \in Z^{+}

and D square free (i.e.

\sqrt{D} \notin Z

), a generalized Pell equation

X^{2} - D Y^{2} = N

admits either no solution, or one or several fundamental solution(s)

(X_{1}, Y_{1})

and also one or several infinite branches of solutions

(X_{n}, Y_{n})

. Several methods exist to find the fundamental solutions of the generalized Pell equation (see [15,19,31]). Two methods are used further: first a brute force search method, i.e. trying several values of Y until the smallest

X_{1} = \sqrt{N + D Y_{1}^{2}} \in Z^{+}

is found; second, Matthews’ method [16] based on an algorithm by Frattini [4,5,6] using Nagell’s bounds [18,21]. Once fundamental solution(s)

(X_{1}, Y_{1})

have been found one way or another, noting

(x_{f}, y_{f})

the fundamental solutions of the related simple Pell equation

X^{2} - D Y^{2} = 1

, the other solutions

(X_{n}, Y_{n})

can be found by

X_{n} + \sqrt{D} Y_{n} = \pm (X_{1} + \sqrt{D} Y_{1}) {(x_{f} + \sqrt{D} y_{f})}^{n}

(24)

for a proper choice of sign ± [17], which can be written also in function of Chebyshev’s polynomials [28]

\begin{matrix} X_{n} & = & X_{1} T_{n - 1} (x_{f}) + D Y_{1} y_{f} U_{n - 2} (x_{f}) \end{matrix}

(25)

\begin{matrix} Y_{n} & = & X_{1} y_{f} U_{n - 2} (x_{f}) + Y_{1} T_{n - 1} (x_{f}) \end{matrix}

(26)

where

T_{n - 1} (x_{f})

and

U_{n - 2} (x_{f})

are Chebyshev polynomials of the first and second kinds evaluated at

x_{f}

The generalized Pell equation (17) can be written as

{(λ f_{μ, 0})}^{2} - (\frac{λ^{2} M_{μ, 0}}{δ^{2}}) {(η + δ)}^{2} = \frac{λ^{2} (M_{μ, 0} + 1)}{3}

(27)

with

X = λ f_{μ, 0}

Y = (η + δ)

D = (λ^{2} M_{μ, 0} / δ^{2})

and

N = λ^{2} (M_{μ, 0} + 1) / 3

, and where

λ = 1

δ \equiv 1

5 (m o d 6)

and

λ = 3

δ \equiv 0 (m o d 6)

To use Matthews’ method [16], the parameters D and N must be fixed with values of

δ

and

M_{μ, 0}

that can be chosen from the allowed congruent values (see Section 3) and be tried one by one until fundamental solutions are found. Alternatively, fixing the values of

η

and

δ

, a brute force search method can be used to find

f_{μ, 0} \in Z^{+}

for the smallest value of

M_{μ, 0} \in Z^{+}

, with from (11)

f_{μ, 0} = \sqrt{\frac{M_{μ, 0} (3 {(η + δ)}^{2} + δ^{2}) + δ^{2}}{3 δ^{2}}}

(28)

Relation (28) yields then the allowed values of

μ, M_{μ, 0}

and

f_{μ, 0}

given in Table 2for

δ = 1,

μ = η \in Z^{+}

, for

0 \leq μ \leq 100

(1)

and in [29] for

δ \neq 1,

μ = (η / δ) \in Q^{+}

, for

0 < η, δ \leq 100

Once a set of values has been found for

η, δ, M_{μ, 0}

and

f_{μ, 0}

as fundamental solution(s) of the generalized Pell equation (27), other allowed values for

η

and

f_{μ, 0}

can be found from the other solutions of (27) using the values of

δ

and

M_{μ, 0}

by (25) and (26) written as

\begin{matrix} f_{μ_{n}, 0} & = & f_{μ_{1}, 0} T_{n - 1} (x_{f}) + (\frac{λ M_{μ, 0}}{δ^{2}}) (η_{1} + δ) y_{f} U_{n - 2} (x_{f}) \end{matrix}

(29)

\begin{matrix} η_{n} & = & λ f_{μ_{1}, 0} y_{f} U_{n - 2} (x_{f}) + (η_{1} + δ) T_{n - 1} (x_{f}) - δ \end{matrix}

(30)

Example 1.

For

δ = 1

and

μ = η = 1

M_{1, 0} = 2

and

f_{1, 0} = 3

from Table 1. Using

M_{1, 0}

and δ as constants in (27) with

λ = 1

, it reduces to a simple Pell equation

f_{μ, 0}^{2} - 2 {(μ + 1)}^{2} = 1

(see e.g. [7,9,23,25]) which admits the single fundamental solution

(X_{1}, Y_{1}) = (f_{μ_{1}, 0}, (μ_{1} + 1)) = (3, 2)

(f_{μ_{1}, 0}, μ_{1}) = (3, 1)

and an infinity of other solutions that can be found

\forall n \in Z^{+}

\begin{matrix} f_{μ_{n}, 0} & = & \frac{{(3 + 2 \sqrt{2})}^{n} + {(3 - 2 \sqrt{2})}^{n}}{2} = 3, 17, 99, 577, 3363, . . . \end{matrix}

(31)

\begin{matrix} μ_{n} & = & \frac{{(3 + 2 \sqrt{2})}^{n} - {(3 - 2 \sqrt{2})}^{n}}{2 \sqrt{2}} - 1 = 1, 11, 69, 407, 2377, . . . \end{matrix}

(32)

where

μ_{n}

are the Pell numbers [32] of even indices minus one. These new values of

(f_{μ_{n}, 0}, μ_{n})

for

n > 1

define new groupings around straight lines of general equation

μ_{n} M = 2 a + c_{μ_{n}}

, with the initial value

M_{μ_{n}, 0} = 2

For

δ = 1

η = 1

M_{1, 0} = 2

f_{1, 0} = 3

ν = 1

, (19) to (21) yield

M_{1, k} = 39 k^{2} + 18 k + 2

a_{1, 1, k} = (39 k^{2} + 5 k) / 2

a_{2, 1, k} = (39 k^{2} + 31 k + 6) / 2

s_{1, 1, k} = (507 k^{3} + 273 k^{2} + 44 k + 2) / 2

s_{2, 1, k} = (507 k^{3} + 429 k^{2} + 116 k + 10) / 2

and values of

M_{μ, k}

a_{1, μ, k}

and

a_{2, μ, k}

for

- 10 \leq k \leq 10

are given in Table 2.

Table 2. Values of

M_{μ, k}

a_{1, μ, k}

a_{2, μ, k}

for

μ = 1, 5, (1 / 6)

and

- 10 \leq k \leq 10

Table 2. Values of

M_{μ, k}

a_{1, μ, k}

a_{2, μ, k}

for

μ = 1, 5, (1 / 6)

and

- 10 \leq k \leq 10

	$μ = 1$			$μ = 5$			$μ = 1 / 6$
k	$M_{1, k}$	$a_{1, 1, k}$	$a_{2, 1, k}$	$M_{5, k}$	$a_{1, 5, k}$	$a_{2, 5, k}$	$M_{1 / 6, k}$	$a_{1, 1 / 6, k}$	$a_{2, 1 / 6, k}$
0	2	0	3	11	18	38	312	15	38
-1	23	17	7	218	590	501	5784	532	433
1	59	22	38	458	1081	1210	12408	962	1107
-2	122	73	50	1079	2797	2599	28824	2513	2292
2	194	83	112	1559	3779	4017	42072	3373	3640
-3	299	168	132	2594	6639	6332	69432	5958	5615
3	407	183	225	3314	8112	8459	89304	7248	7637
-4	554	302	253	4763	12116	11700	127608	10867	10402
4	698	322	377	5723	14080	14536	154104	12587	13098
-5	887	475	413	7586	19228	18703	203352	17240	16653
5	1067	500	568	8786	21683	22248	236472	19390	20023
-6	1298	687	612	11063	27975	27341	296664	25077	24368
6	1514	717	798	12503	30921	31595	336408	27657	28412
-7	1787	938	850	15194	38357	37614	407544	34378	33547
7	2039	973	1067	16874	41794	42577	453912	37388	38265
-8	2354	1228	1127	19979	50374	49522	535992	45143	44190
8	2642	1268	1375	21899	54302	55194	588984	48583	49582
-9	2999	1557	1443	25418	64026	63065	682008	57372	56297
9	3323	1602	1722	27578	68445	69446	741624	61242	62363
-10	3722	1925	1798	31511	79313	78243	845592	71065	69868
10	4082	1975	2108	33911	84223	85333	911832	75365	76608

Example 2.

For

δ = 1

and

μ = η = 5

M_{5, 0} = 11

and

f_{5, 0} = 20

from Table 1. Using

M_{5, 0}

and δ as constants in (27) with

λ = 1

yield the generalized Pell equation

f_{μ, 0}^{2} - 11 {(μ + 1)}^{2} = 4

. Using Matthews’ method [16] yields the single fundamental solution

(f_{μ_{1}, 0}, (μ_{1} + 1)) = (2, 0)

which is of no use. However, as the right hand term is a squared integer, the equation can be rewritten as a simple Pell equation

{(f_{μ, 0} / 2)}^{2} - 11 {((μ + 1) / 2)}^{2} = 1

, which admits the fundamental solution

((f_{μ_{1}, 0} / 2), ((μ_{1} + 1) / 2)) = (10, 3)

(f_{μ_{1}, 0}, μ_{1}) = (20, 5)

and an infinity of other solutions

\forall n \in Z^{+}

\begin{matrix} f_{μ_{n}, 0} & = & {(10 + 3 \sqrt{11})}^{n} + {(10 - 3 \sqrt{11})}^{n} = 20, 398, 7940, 158402, . . . \end{matrix}

(33)

\begin{matrix} μ_{n} & = & \frac{{(10 + 3 \sqrt{11})}^{n} - {(10 - 3 \sqrt{11})}^{n}}{\sqrt{11}} - 1 = 5, 119, 2393, 47759, . . . \end{matrix}

(34)

For

δ = 1

η = 5

M_{5, 0} = 11

f_{5, 0} = 20

ν = 1

, (19) to (21) yield (see Table 2)

M_{5, k} = 327 k^{2} + 120 k + 11

a_{1, 5, k} = (1635 k^{2} + 491 k + 36) / 2

a_{2, 5, k} = (1635 k^{2} + 709 k + 76) / 2

s_{1, 5, k} = (35643 k^{3} + 17658 k^{2} + 2879 k + 154) / 2

s_{2, 5, k} = (35643 k^{3} + 21582 k^{2} + 4319 k + 286) / 2

Example 3.

For

η = 1

and

δ = 6

M_{1 / 6, 0} = 312

and

f_{1 / 6, 0} = 23

[29]. Using

M_{1 / 6, 0}

and δ as constants in (27) with

λ = 3

yields

{(3 f_{μ, 0})}^{2} - 78 {(η + 6)}^{2} = 939

, which by [16] has two fundamental solutions

((3 f_{μ_{1}, 0}), (η_{1} + 6)) = (69, 7)

and

(381, 43)

, yielding

(f_{μ_{1}, 0}, η_{1}) = (23, 1)

and

(127, 37)

. The fundamental solutions of the related simple Pell equation

X^{2} - 78 Y^{2} = 1

are

(x_{f}, y_{f}) = (53, 6)

. Other values of

(f_{μ_{n}, 0}, η_{n})

can be found on the two infinite branches corresponding to these two fundamental solutions by (29) and (30) as

\begin{matrix} f_{μ_{n}, 0} & = & 23 T_{n - 1} (53) + 1092 U_{n - 2} (53) \end{matrix}

(35)

\begin{matrix} = & 23, 2311, 244943, 25961647, 2751689639, . . . \end{matrix}

(36)

\begin{matrix} η_{n} & = & 414 U_{n - 2} (53) + 7 T_{n - 1} (53) - 6 \end{matrix}

(37)

\begin{matrix} = & 1, 779, 83197, 8818727, 934702489, . . . \end{matrix}

(38)

for the first fundamental solution, and

\begin{matrix} f_{μ_{n}, 0} & = & 127 T_{n - 1} (53) + 6708 U_{n - 2} (53) \end{matrix}

(39)

\begin{matrix} = & 127, 13439, 1424407, 150973703, 16001788111, . . . \end{matrix}

(40)

\begin{matrix} η_{n} & = & 2286 U_{n - 2} (53) + 43 T_{n - 1} (53) - 6 \end{matrix}

(41)

\begin{matrix} = & 37, 4559, 483841, 51283211, 5435537149, . . . \end{matrix}

(42)

for the second fundamental solution. For

η = 1

δ = 6

M_{1 / 6, 0} = 312

f_{1 / 6, 0} = 23

ν = (2 / 3)

, (19) to (21) yield (see Table 2)

M_{1 / 6, k} = 8784 k^{2} + 3312 k + 312

a_{1, 1 / 6, k} = 732 k^{2} + 215 k + 15

a_{2, 1 / 6, k} = 732 k^{2} + 337 k + 38

s_{1, 1 / 6, k} = 535824 k^{3} + 297924 k^{2} + 55188 k + 3406

s_{2, 1 / 6, k} = 535824 k^{3} + 308172 k^{2} + 59052 k + 3770

5. Conclusions

It is shown that regular linear features exist in the distribution of couples of values a and M in the

(a, M)

plot, where a and M are the first term and the number of terms in sums of consecutive squared integers equal to integer squares. These regular features correspond to groupings of pairs of a values for successive same values of M around straight lines of equation

μ M \approx 2 a

for positive rational values of

μ = (η / δ)

For allowed values of

η

and

δ

such as

η \equiv 1 (m o d 2)

and

δ \equiv 0, 1

5 (m o d 6)

, if

M_{μ, k} = δ^{2} (3 {(a_{2, μ, k} - a_{1, μ, k})}^{2} - 1) / (3 {(η + δ)}^{2} + δ^{2})

holds

\forall k \in Z

and for pairs

(a_{1, μ, k}, a_{2, μ, k})

, then the sums of

M_{μ, k}

consecutive squared integers starting with

a_{1, μ, k}

a_{2, μ, k}

are always equal to squared integers

s_{1, μ, k}^{2}

s_{2, μ, k}^{2}

\forall k \in Z

. Parametric equations are found in function of

k \in Z

: linear for

(a_{2, μ, k} - a_{1, μ, k})

, quadratic for

M_{μ, k}

a_{1, μ, k}

and

a_{2, μ, k}

, and cubic for

s_{1, μ, k}

and

s_{2, μ, k}

The allowed values of

η, δ, M_{μ, 0}

and of the difference

f_{μ, 0} = a_{2, μ, 0} - a_{1, μ, 0}

are found by solving the generalized Pell equation

{(δ f_{μ, 0})}^{2} - M_{μ, 0} {(η + δ)}^{2} = δ^{2} (M_{μ, 0} + 1) / 3

and further allowed values of

η_{n}

and

f_{μ_{n}, 0}

can be calculated for fixed values of

δ

and

M_{μ, 0}

using Chebyshev polynomials.

Notes

1	[22] gives all values of $(μ + 1)$ such that $(3 {(μ + 1)}^{2} + 1)$ is prime for which (28) holds.

References

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W.S. Anglin, The Square Pyramid Puzzle, American Mathematical Monthly, 97, 120-124, 1990.
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M. Laub, Squares Expressible as a Sum of n Consecutive Squares, Advanced Problem 6552, American Mathematical Monthly, 97, 622-625, 1990.
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Figure 1. Distribution of M versus a for the 4078 couples

(a, M)

found such as (1) holds, with

1 \leq a \leq 10^{5}

and

2 \leq M \leq 10^{5}

Figure 1. Distribution of M versus a for the 4078 couples

(a, M)

found such as (1) holds, with

1 \leq a \leq 10^{5}

and

2 \leq M \leq 10^{5}

Table 1. Values of

M_{μ, 0}

and

f_{μ, 0}

for

0 \leq μ \leq 100

Table 1. Values of

M_{μ, 0}

and

f_{μ, 0}

for

0 \leq μ \leq 100

$μ$	$M_{μ, 0}$	$f_{μ, 0}$	$μ$	$M_{μ, 0}$	$f_{μ, 0}$	$μ$	$M_{μ, 0}$	$f_{μ, 0}$
1	2	3	29	26	153	63	3263	3656
5	11	20	33	299	588	67	9563	6650
7	74	69	35	479	788	69	2	99
11	2	17	39	1391	1492	77	74	671
15	194	223	43	59	338	81	1202	2843
19	122	221	49	491	1108	83	146	1015
21	983	690	53	1739	2252	85	1874	3723
25	506	585	55	383	1096	97	23	470
27	47	192	57	2327	2798

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Linear Relations between Numbers of Terms and First Terms of Sums of Consecutive Squared Integers Equal to Squared Integers

Abstract

1. Introduction

2. Linear features in the a , M plot

3. Parametric expressions of f μ , k , M μ , k , a j , μ , k , s j , μ , k

4. Finding allowed values of μ = η / δ and M μ , k

5. Conclusions

Notes

References

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2. Linear features in the $(a, M)$ plot

3. Parametric expressions of $f_{μ, k}$ , $M_{μ, k}$ , $a_{j, μ, k}$ , $s_{j, μ, k}$

4. Finding allowed values of $μ = (η / δ)$ and $M_{μ, k}$