Infinite families of congruences for $2$ and $13$-core partitions

TL;DR

This paper establishes infinite families of congruences and identities for 2-core and 13-core partition functions modulo 2, extending previous results and employing Hecke eigenform theory.

Contribution

It introduces new infinite families of congruences for $a_2(n)$ and $a_{13}(n)$ modulo 2 using Hecke eigenform theory, generalizing prior work.

Findings

Derived infinite families of congruences for $a_2(n)$ and $a_{13}(n)$ modulo 2.

Established multiplicative identities for these core partition functions.

Extended previous results by Das on similar congruences.

Abstract

A partition of $n$ is called a $t$-core partition if none of its hook number is divisible by $t.$ In 2019, Hirschhorn and Sellers \cite{Hirs2019} obtained a parity result for $3$-core partition function $a_3(n)$. Motivated by this result, both the authors \cite{MeherJindal2022} recently proved that for a non-negative integer $\alpha,$ $a_{3^{\alpha} m}(n)$ is almost always divisible by arbitrary power of $2$ and $3$ and $a_{t}(n)$ is almost always divisible by arbitrary power of $p_i^j,$ where $j$ is a fixed positive integer and $t= p_1^{a_1}p_2^{a_2}\ldots p_m^{a_m}$ with primes $p_i \geq 5.$ In this article, by using Hecke eigenform theory, we obtain infinite families of congruences and multiplicative identities for $a_2(n)$ and $a_{13}(n)$ modulo $2$ which generalizes some results of Das \cite{Das2016}.

Equations

$$\sum _{n=0}^{\infty }{a}_t(n)q^n=\prod _{n=1}^{\infty }\frac{(1-q^{tn}{)}^t}{(1-q^n)}=\frac{(q^t;q^t{)}_{\infty }^t}{(q;q{)}_{\infty }},$$

$$a_p(p^{j}n-{\delta }_p)\equiv 0(\mathrm{mod}{p}^j)$$

$$a_{5^j}(5^{j}n-{\delta }_{5,j})$$

$$a_{7^j}(7^{j}n-{\delta }_{7,j})$$

$$a_{11^j}(11^{j}n-{\delta }_{11,j})$$

$$a_3(n)=\{\begin{array}{ll}1(\mathrm{mod}2)& \text{if }n=3r^2+2r\text{ for some integer }r,\\ 0(\mathrm{mod}2)& \text{otherwise.}\end{array}$$

$${ϵ}_p=\{\begin{array}{ll}1& \text{ if }p\not\equiv 1(\mathrm{mod}8),\\ 8& \text{ if }p\equiv 1(\mathrm{mod}8).\end{array}$$

$$\Gamma :=\mathrm{S}{\mathrm{L}}_2(\mathbb{Z})=$$

$${\Gamma }_{\infty }:=$$

$${\Gamma }_0(N):=$$

$${\Gamma }_1(N):=$$

$$\Gamma (N):=\left\{\left[\begin{array}{ccc}a& & b\\ c& & d\end{array}\right]\in \mathrm{S}{\mathrm{L}}_2(\mathbb{Z}):a\equiv d\equiv 1(\mathrm{mod}N),b\equiv c\equiv 0(\mathrm{mod}N)\right\}.$$

$$[\Gamma :{\Gamma }_0(N)]=N\prod _{p|N}\left(1+\frac{1}{p}\right)$$

$$\mathrm{G}{\mathrm{L}}_2^{+}(\mathbb{R}):=\left\{\left[\begin{array}{ccc}a& & b\\ c& & d\end{array}\right]:a,b,c,d\in \mathbb{R},ad-bc>0\right\}$$

$$(f{|}_k\gamma )(z):=(\det \gamma {)}^{k/2}(cz+d{)}^{-k}f(\gamma z).$$

$$f\left(\frac{az+b}{cz+d}\right)=(cz+d{)}^{k}f(z).$$

$$(f{|}_k\gamma )(z):=\sum _{n\ge 0}{a}_{\gamma }(n)q_N^n$$

$$f\left(\frac{az+b}{cz+d}\right)=\chi (d)(cz+d{)}^{k}f(z)$$

$$\eta (z):=q^{1/24}(q;q{)}_{\infty }=q^{1/24}\prod _{n=1}^{\infty }(1-q^n)$$

$$f(z):=\prod _{\delta |N}\eta (\delta z{)}^{r_{\delta }}$$

$$\sum _{\delta |N}\delta r_{\delta }\equiv 0(\mathrm{mod}24)\text{and}\sum _{\delta |N}\frac{N}{\delta }{r}_{\delta }\equiv 0(\mathrm{mod}24),$$

$$f\left(\frac{az+b}{cz+d}\right)=\chi (d)(cz+d{)}^{k}f(z)$$

$$\frac{N}{24}\sum _{\delta |N}\frac{\gcd (d,\delta {)}^2{r}_{\delta }}{\gcd (d,\frac{N}{d})d\delta }.$$

$$f(z)|T_m:=\sum _{n=0}^{\infty }\left(\sum _{d|\gcd (n,m)}\chi (d)d^{k-1}a\left(\frac{mn}{d^2}\right)\right)q^n.$$

$$f(z)|T_p:=\sum _{n=0}^{\infty }\left(a(pn)+\chi (p)p^{k-1}a\left(\frac{n}{p}\right)\right)q^n.$$

$$\sum _{n=1}^{\infty }b(n)q^n=q(q^8;q^8{)}_{\infty }(q^{16};q^{16}{)}_{\infty }\text{and}\sum _{n=0}^{\infty }c(n)q^n:=(q;q{)}_{\infty }^3.$$

$$b(pn)=(-1)\left(\frac{-2}{p}\right)b\left(\frac{n}{p}\right).$$

$$b(p^{2}n+pj)=0.$$

$$\sum _{n=1}^{\infty }b(n)q^n=\eta (8z)\eta (16z).$$

$$\sum _{n=1}^{\infty }b(n)q^n=\eta (8z)\eta (16z)=q-q^9-2q^{17}+\cdots .$$

$$\eta (8z)\eta (16z)|T_p=\sum _{n=1}^{\infty }\left(b(pn)+\left(\frac{-128}{p}\right)b\left(\frac{n}{p}\right)\right)q^n=\lambda (p)\sum _{n=1}^{\infty }b(n)q^n.$$

$$b(pn)+\left(\frac{-2}{p}\right)b\left(\frac{n}{p}\right)=\lambda (p)b(n).$$

Full text

2010 Mathematics Subject Classification: Primary 11P83, Secondary 11F11

Keywords: $t$-core partitions; Eta-quotients; Congruence; modular forms.

1. Introduction

A partition $\beta=(\beta_{1},\beta_{2},\cdots,\beta_{r})$ of $n$ is a non-increasing sequence of positive integers whose sum is $n$ and the positive integers $\beta_{i}$ are called parts of the partition $\beta$. A partition $\beta$ of $n$ can be represented by the Young diagram $[\beta]$ (also known as the Ferrers graph) which consists of the $s$ number of rows such that the $i^{th}$ row has $\beta_{i}$ number of dots $\bullet$ and all the rows start in the first column. An illustration of the Young diagram for $\beta=(\beta_{1},\beta_{2},\cdots,\beta_{r})$ is as follows.

$[\beta]$:=

$\bullet$ $\bullet$ \cdots$$\cdots$$\cdots   $\bullet$ $\beta_{1}$ dots

$\bullet$ $\bullet$ \cdots$$\cdots   $\bullet$ $\beta_{2}$ dots

$\vdots$

⋮

$\bullet$ $\bullet$ $\cdots$   $\bullet$ $\beta_{r}$ dots

For $1\leq i\leqslant r$ and $1\leq j\leq\beta_{i}$, the dot of $[\beta]$ which lies in the $i^{th}$ row and $j^{th}$ column is denoted by $(i,j)^{th}$-dot of $\beta$. Let $\beta_{j}^{{}^{\prime}}$ denote the number of dots in $j^{th}$ column. The hook number $H_{i,j}$ of $(i,j)^{th}$-dot is defined by $\beta_{i}+\beta_{j}^{{}^{\prime}}-i-j+1$. In other words, $H_{i,j}=1+h_{0}$ where $h_{0}$ is the sum of the number of dots lying right to the $(i,j)^{th}$-dot in the $i^{th}$ row, the number of dots lying below the $(i,j)^{th}$-dot in the $j^{th}$ column. Given a partition $\beta$ of $n$, we say that it is a $t$-core partition if none of its hook number is divisible by $t$.

Example 1. The Young diagram of the partition $\beta=(6,3,1)$ of $10$ is

[TABLE]

where the superscript on each dot represents its hook number. It can be easily observed that this is a $t$-core partition of $10$ for $t\in\{7\}$ and $t\geqslant 9$.

Example 2. There are no $3$-core partitions of $7$. This can be easily verified by looking at the Young diagram of each partition of $7$.

For a positive integer $n$, let $a_{t}(n)$ denote the number of $t$-core partitions of $n.$ Its generating function is given by

[TABLE]

where $(a;q)_{\infty}=(1-a)(1-aq)(1-aq^{2})\cdots$.

In [4, Corollary 1], Garvan, Kim, Stanton obtained the congruence

[TABLE]

where $p\in\{5,7,11\}$, $n$, $j$ are positive integers and $\delta_{p}=\frac{p^{2}-1}{24}$. In [6, Proposition 3], Granville and Ono proved similar congruences, namely

[TABLE]

where $n$, $j$ are positive integers and $\delta_{p,j}\equiv\frac{1}{24}\pmod{p^{j}}$ for $p\in\{5,7,11\}$.

In 2019, Hirschhorn and Sellers [7] proved a parity result for $a_{3}(n)$, i.e. for all $n\geq 0$,

[TABLE]

Motivated by this result, both the authors proved that for a non-negative integer $\alpha,$ $a_{3^{\alpha}m}(n)$ is almost always divisible by arbitrary power of $2$ and $3.$ Moreover, they also proved that $a_{t}(n)$ is almost always divisible by arbitrary power of $p_{i}^{j},$ where $j$ is a fixed positive integer and $t=p_{1}^{a_{1}}p_{2}^{a_{2}}\ldots p_{m}^{a_{m}}$ with primes $p_{i}\geq 5.$ In this following theorem, we obtain infinite families of congruences modulo $2$ for $a_{2}(n)$ and $a_{13}(n)$ by using Hecke eigen form theory.

Theorem 1.1**.**

Let $k$ and $n$ be non-negative integers. For each $1\leq i\leq k+1,$ let $p_{1},p_{2},\ldots,p_{k+1}$ be prime numbers such that $p_{i}\geq 5$. Then for any integer $j\not\equiv 0\pmod{p_{k+1}},$ we have

• (i)

$a_{2}\left(p_{1}^{2}p_{2}^{2}\cdots p_{k+1}^{2}n+\frac{p_{1}^{2}p_{2}^{2}\cdots p_{k}^{2}p_{k+1}\left(8j+p_{k+1}\right)-1}{8}\right)\equiv 0\pmod{2}.$**

• (ii)

$a_{13}\left(104p_{1}^{2}p_{2}^{2}\cdots p_{k+1}^{2}n+13p_{1}^{2}p_{2}^{2}\cdots p_{k}^{2}p_{k+1}\left(\epsilon_{p}j+p_{k+1}\right)-7\right)\equiv 0\pmod{2}$* where*

[TABLE]

Corollary 1.1**.**

Let $n$ and $k$ be non-negative integers. For a prime $p\geq 5$ and an integer $j\not\equiv 0\pmod{p}$, we have

• (i)

$a_{2}\left(p^{2(k+1)}n+p^{2k+1}j+\frac{p^{2k+2}-1}{8}\right)\equiv 0\pmod{2}.$**

• (ii)

$a_{13}\left(104p^{2k+2}n+13\epsilon_{p}p^{2k+1}j+13p^{2k+2}-7\right)\equiv 0\pmod{2}.$**

Furthermore, we prove the following multiplicative formulae for $2$-core partitions and $13$-core partitions modulo $2$.

Theorem 1.2**.**

Let $k$ be a positive integer and $p$ be a prime number such that $p\equiv 7\pmod{8}.$ Let $r$ be a non-negative integer such that $p$ divides $8r+7,$ then

• (i)

$a_{2}\left(p^{k+1}n+pr+\frac{7p-1}{8}\right)\equiv(-1)\left(\frac{-2}{p}\right)a_{2}\left(p^{k-1}n+\frac{8r+7-p}{8p}\right)\pmod{2}.$**

• (ii)

$a_{13}\left(104p^{k+1}n+104pr+91p-7\right)\equiv(-1)\left(\frac{-2}{p}\right)a_{13}\left(104p^{k-1}n+\frac{104r+91}{p}-7\right)\pmod{2}.$**

Corollary 1.2**.**

Let $k$ be a positive integer and $p$ be a prime number such that $p\equiv 7\pmod{8}.$ Then

• (i)

$a_{2}\left(p^{2k}n+\frac{p^{2k}-1}{8}\right)\equiv\left(-1\right)^{k}\left(\frac{-2}{p}\right)^{k}a_{2}(n)\pmod{2}.$**

• (ii)

$a_{13}\left(104p^{2k}n+13p^{2k}-7\right)\equiv\left(-1\right)^{k}\left(\frac{-2}{p}\right)^{k}a_{13}(104n+6)\pmod{2}.$**

2. Preliminaries

We recall some basic facts and definition on modular forms. For more details, we refer to [8, 12]. We start with some matrix groups. We define

[TABLE]

For a positive integer $N$, we define

[TABLE]

and

[TABLE]

A subgroup $\Gamma$ of $\mathrm{SL_{2}}(\mathbb{Z})$ is called a congruence subgroup if it contains $\Gamma(N)$ for some positive integer $N$ and the smallest $N$ with this property is called its level. Note that $\Gamma_{0}(N)$ and $\Gamma_{1}(N)$ are congruence subgroups of level $N,$ whereas $\mathrm{SL_{2}}(\mathbb{Z})$ and $\Gamma_{\infty}$ are congruence subgroups of level $1.$ The index of $\Gamma_{0}(N)$ in $\Gamma$ is

[TABLE]

where $p$ runs over prime divisors of $N$.

Let $\mathbb{H}$ denote the upper half of the complex plane. The group

[TABLE]

acts on $\mathbb{H}$ by $\begin{bmatrix}a&&b\\ c&&d\end{bmatrix}z=\frac{az+b}{cz+d}.$ We identify $\infty$ with $\frac{1}{0}$ and define $\begin{bmatrix}a&&b\\ c&&d\end{bmatrix}\frac{r}{s}=\frac{ar+bs}{cr+ds},$ where $\frac{r}{s}\in\mathbb{Q}\cup\{\infty\}$. This gives an action of $\mathrm{GL_{2}^{+}}(\mathbb{R})$ on the extended half plane $\mathbb{H}^{*}=\mathbb{H}\cup\mathbb{Q}\cup\{\infty\}$. Suppose that $\Gamma$ is a congruence subgroup of $\mathrm{SL_{2}}(\mathbb{Z})$. A cusp of $\Gamma$ is an equivalence class in $\mathbb{P}^{1}=\mathbb{Q}\cup\{\infty\}$ under the action of $\Gamma$.

The group $\mathrm{GL_{2}^{+}}(\mathbb{R})$ also acts on functions $f:\mathbb{H}\rightarrow\mathbb{C}$. In particular, suppose that $\gamma=\begin{bmatrix}a&&b\\ c&&d\end{bmatrix}\in\mathrm{GL_{2}^{+}}(\mathbb{R})$. If $f(z)$ is a meromorphic function on $\mathbb{H}$ and $k$ is an integer, then define the slash operator $|_{k}$ by

[TABLE]

Definition 2.1**.**

Let $\Gamma$ be a congruence subgroup of level $N$. A holomorphic function $f:\mathbb{H}\rightarrow\mathbb{C}$ is called a modular form of integer weight $k$ on $\Gamma$ if the following hold:

1. $(1)$

For all $z\in\mathbb{H}$ and $\begin{bmatrix}a&&b\\ c&&d\end{bmatrix}\in\Gamma$,

[TABLE] 2. $(2)$

If $\gamma\in SL_{2}(\mathbb{Z})$, then $(f|_{k}\gamma)(z)$ has a Fourier expnasion of the form

[TABLE]

where $q_{N}:=e^{2\pi iz/N}$.

For a positive integer $k$, the complex vector space of modular forms of weight $k$ with respect to a congruence subgroup $\Gamma$ is denoted by $M_{k}(\Gamma)$.

Definition 2.2**.**

[12, Definition 1.15]** Let $\chi$ be a Dirichlet character modulo $N$. We say that a modular form $f\in M_{k}(\Gamma_{1}(N))$ has Nobentypus character $\chi$ if

[TABLE]

for all $z\in\mathbb{H}$ and $\begin{bmatrix}a&&b\\ c&&d\end{bmatrix}\in\Gamma_{0}(N)$. The space of such modular forms is denoted by $M_{k}(\Gamma_{0}(N),\chi)$.

The relevant modular forms for the results obtained in this article arise from eta-quotients. We recall the Dedekind eta-function $\eta(z)$ which is defined by

[TABLE]

where $q:=e^{2\pi iz}$ and $z\in\mathbb{H}$. A function $f(z)$ is called an eta-quotient if it is of the form

[TABLE]

where $N$ and $r_{\delta}$ are integers with $N>0$.

Theorem 2.1**.**

[12, Theorem 1.64]** If $f(z)=\prod\limits_{\delta|N}\eta(\delta z)^{r_{\delta}}$ is an eta-quotient such that $k=\frac{1}{2}$ $\sum_{\delta|N}r_{\delta}\in\mathbb{Z}$,

[TABLE]

then $f(z)$ satisfies

[TABLE]

for each $\begin{bmatrix}a&&b\\ c&&d\end{bmatrix}\in\Gamma_{0}(N)$. Here the character $\chi$ is defined by $\chi(d):=\left(\frac{(-1)^{k}s}{d}\right)$ where $s=\prod_{\delta|N}\delta^{r_{\delta}}$.

Theorem 2.2**.**

[12, Theorem 1.65]** Let $c,d$ and $N$ be positive integers with $d|N$ and $\gcd(c,d)=1$. If $f$ is an eta-quotient satisfying the conditions of Theorem 2.1 for $N$, then the order of vanishing of $f(z)$ at the cusp $\frac{c}{d}$ is

[TABLE]

Suppose that $f(z)$ is an eta-quotient satisfying the conditions of Theorem 2.1 and that the associated weight $k$ is a positive integer. If $f(z)$ is holomorphic at all of the cusps of $\Gamma_{0}(N)$, then $f(z)\in M_{k}(\Gamma_{0}(N),\chi)$. Theorem 2.2 gives the necessary criterion for determining orders of an eta-quotient at cusps. In the proofs of our results, we use Theorems 2.1 and 2.2 to prove that $f(z)\in M_{k}(\Gamma_{0}(N),\chi)$ for certain eta-quotients $f(z)$ we consider in the sequel.

We recall the definition of Hecke operators and a few relevant results. Let $m$ be a positive integer and $f(z)=\sum\limits_{n=0}^{\infty}a(n)q^{n}\in M_{k}(\Gamma_{0}(N),\chi)$. Then the action of Hecke operator $T_{m}$ on $f(z)$ is defined by

[TABLE]

In particular, if $m=p$ is a prime, we have

[TABLE]

We note that $a(n)=0$ unless $n$ is a non-negative integer.

3. Proofs of Theorem 1.1 and 1.2

3.1. Prelude to the proofs

We define

[TABLE]

If $p\nmid n$, then we set $b\left(\frac{n}{p}\right)=0$ and $c\left(\frac{n}{p}\right)=0$. We have the following result.

Lemma 3.2**.**

*For $n\geq 0$ and for a prime $p\not\equiv 1\pmod{8}$, we have *

[TABLE]

*Further if $j\not\equiv 0\pmod{p}$, then *

[TABLE]

Proof.

Let $p$ be a prime with $p\not\equiv 1\pmod{8}$. Using (2.1), we note that

[TABLE]

By using Theorem 2.1, we obtain that $\eta(8z)\eta(16z)\in S_{1}(\Gamma_{0}(128),\left(\frac{-128}{\bullet}\right)).$ Thus $\eta(8z)\eta(16z)$ has the Fourier expansion given by

[TABLE]

Therefore, $b(n)=0$ for all $n\geq 0$ with $n\not\equiv 1\pmod{8}$. Since $\eta(8z)\eta(16z)$ is a Hecke eigenform, we obtain from [9, Table 1] that

[TABLE]

Note that $\left(\frac{-128}{p}\right)=\left(\frac{-2}{p}\right)$. Comparing the coefficients of $q^{n}$ on both sides of the above equation, we get

[TABLE]

Since $b(1)=1$ and $b(\frac{1}{p})=0,$ by substituting $n=1$ in the above expression, we get $b(p)=\lambda(p).$ Further, since $b(p)=0$, we obtain that $\lambda(p)=0$. Hence, we conclude from (3.4) that

[TABLE]

which proves (3.2). For $j\not\equiv 0\pmod{p}$, replacing $n$ by $pn+j$ in (3.5), we get $b(p^{2}n+pj)=0$ which proves (3.3). ∎

Lemma 3.3**.**

*For $n\geq 0$ and for a prime $p\equiv 1\pmod{4}$, we have *

[TABLE]

*If $p\nmid n$, then *

[TABLE]

Proof.

From [2, Page 39, Entry 24(ii)], we have

[TABLE]

Thus

[TABLE]

This implies

[TABLE]

Note that if $(2k+1)^{2}=8pn+p^{2}$, then $p|(2k+1)$ and therefore we can write $2k+1=p(2k^{{}^{\prime}}+1)$ for some positive integer $k^{{}^{\prime}}$. Further for such $k$, we have $k=\frac{2k+1}{2}-\frac{1}{2}=\frac{p(2k^{{}^{\prime}}+1)}{2}-\frac{1}{2}=pk^{{}^{\prime}}+\frac{p-1}{2}$ which gives $(-1)^{k}=(-1)^{k^{{}^{\prime}}}$. Hence

[TABLE]

Replacing $n$ by $pn$, we obtain (3.6). Also, (3.7) follows since $c\left(\frac{n}{p}\right)=0$ if $p\nmid n$. This completes the proof. ∎

We recall the following identity for $13$-core partitions obtained by Kuwali Das.

Lemma 3.4**.**

[3*, Theorem 1]** We have *

[TABLE]

Lemma 3.5**.**

*For $j\not\equiv 0\pmod{p}$ and $n\geq 0$, we have *

[TABLE]

*where *

[TABLE]

Proof.

We consider the two cases $p\not\equiv 1\pmod{8}$ and $p\equiv 1\pmod{8}$ separately as follows.

**Case 1: $p\not\equiv 1\pmod{8}.$

**From (1.1), we have

[TABLE]

Thus using Lemma 3.4, we yield

[TABLE]

From (3.1) and (3.12), we get

[TABLE]

Let $r\not\equiv 0\pmod{p}$. From (3.3), we have

[TABLE]

Replacing $n$ by $8n-pr+1$, we obtain

[TABLE]

Note that $8p^{2}n-p^{3}r+p^{2}+pr=8(p^{2}n-pr\frac{p^{2}-1}{8}+\frac{p^{2}-1}{8})+1$. Therefore using (3.13), we obtain

[TABLE]

Since $\gcd(\frac{p^{2}-1}{8},p)=1$ and $\gcd(p^{2}-1,p)=1,$ when $r$ runs over a residue system excluding the multiples of $p$, so do $\frac{-r(p^{2}-1)}{8}$ and $-r(p^{2}-1)$. Thus for $j\not\equiv 0\pmod{p}$, (3.14) can be written as

[TABLE]

and

[TABLE]

This proves (3.8) and (3.10) in the case of $p\not\equiv 1\pmod{8}$.

Next, substituting $n$ by $8pn+p$ in (3.2), we obtain

[TABLE]

Note that $8p^{2}n+p^{2}=8(p^{2}n+\frac{p^{2}-1}{8})+1$. Therefore using (3.13) in (3.15), we get

[TABLE]

and

[TABLE]

which proves (3.9) and (3.11) in the case of $p\not\equiv 1\pmod{8}$.

**Case 2: $p\equiv 1\pmod{8}.$

**From (1.1), we have

[TABLE]

From Lemma 3.4, we have

[TABLE]

Invoking (3.1), (3.16) and (3.17), we yield

[TABLE]

If $p\nmid n$, then from (3.7) and (3.18), we get

[TABLE]

Next replacing $n$ by $pn+j$ for $j\not\equiv 0\pmod{p}$, we obtain

[TABLE]

which proves (3.8) and (3.10) in the case of $p\equiv 1\pmod{8}$.

Next using (3.6) and (3.18), we get

[TABLE]

which proves (3.9) and (3.11) in the case of $p\equiv 1\pmod{8}$. ∎

3.6. Proof of Theorem 1.1(i)

For $1\leq i\leq k-1$, we note that

[TABLE]

Thus for $1\leq i\leq k-1$, using (3.9) for $p=p_{i}$ we have

[TABLE]

Also from (3.9), we have

[TABLE]

Therefore from the congruences in the above two displays, we get

[TABLE]

Replacing $n$ by $p_{k+1}^{2}n+\frac{p_{k+1}(8j+p_{k+1})-1}{8}$ in the above expression and then using (3.8) for $p=p_{k+1}$, we get

[TABLE]

when $j\not\equiv 0\pmod{p_{k+1}}$. This completes the proof of Theorem 1.1(i).

3.7. Proof of Theorem 1.1(ii)

The proof is similar to the proof of Theorem 1.1(i). For $1\leq i\leq k-1$, we note that

[TABLE]

Thus for $1\leq i\leq k-1$, (3.11) implies

[TABLE]

Also from (3.11), we have

[TABLE]

Therefore from the above two congruences, we get

[TABLE]

Replacing $n$ by $p_{k+1}^{2}n+\frac{p_{k+1}(\epsilon_{p_{k+1}}j+p_{k+1})-1}{8}$ in the above expression and then using (3.10), we get

[TABLE]

when $j\not\equiv 0\pmod{p_{k+1}}$. This completes the proof of Theorem 1.1(ii).

3.8. Proof of Theorem 1.2

For any prime $p\equiv 7\pmod{8}$, we get from (3.2) that

[TABLE]

Let $r\not\equiv 0\pmod{p}$. Replacing $n$ by $8(p^{k}n+r)+7$, we obtain

[TABLE]

which can be rewritten as

[TABLE]

We note here that $\frac{7p-1}{8}$ and $\frac{8r+7-p}{8p}$ are integers. Therefore using (3.13) and (3.19), we get

[TABLE]

and

[TABLE]

3.9. Proof of Corollary 1.2

Let $p$ be a prime such that $p\equiv 7\pmod{8}.$ Choose a non negative integer $r$ such that $8r+7=p^{2k-1}.$ Substituting $k$ by $2k-1$ in (3.20), we obtain

[TABLE]

Substituting $k$ by $2k-1$ in (3.21), we obtain

[TABLE]