The Journal of the Indian Mathematical Society

Hansraj Gupta

Abstract

In 1920, Mac Mahon* gave a method for determining the parity of p(n)-the number of unrestricted partitions of n. He made use of Jacobi's Elliptic products. His results are here obtained in a simpler and more direct way.

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Hansraj Gupta ¹

Affiliations
1 Hoshiarpur, IN

Abstract

In what follows, p(n, m) denotes the number oi partitions of n into exactly m non-zero summands and p₀(n) the number of unrestricted partitions of n.

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Hansraj Gupta ¹

Affiliations
1 Government College, Hoshiarpur, IN

Abstract

Proceeding on the lines of my paper (1) on the "Congruence Properties of τ(n)", it can be shown that for any odd prime p,

σ (P) = P+I (mod 8)

= σ(P) (mod 8),

where τ(n) is Ramanujan's function.

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Hansraj Gupta ¹

Affiliations
1 Government College, Hoshiarpur, IN

Abstract

A sequence of numbers a_l= a_2, a_3, • a_n,••• is said to be in Medial Progression (M.P.)

if a_r = a_{r_1} +a_r-2 [r≥3]. … (1)

Throughout this paper which develops properties of such a progression, the numbers a_n will be supposed to be positive or negative integers.

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Hansraj Gupta ¹

Affiliations
1 Hoshiarpur, IN

Abstract

In what follows, all small letters denote integers ≥ o; the letter m is used to denote only quadratfrei numbers; p denotes any prime; p_v the vth prime with p₁ = 2. Q(p_v) denotes the set of quadratfrei integers none of whose prime factors exceeds p_v.

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Hansraj Gupta ¹

Affiliations
1 Govt. College, Hoshiarpur, IN

Abstract

If p_k(n) denotes the number of partitions of n into k non-zero summands, then I have shown that for all positive integral values of n ≥ k.

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Hansraj Gupta ¹

Affiliations
1 Hoshiarpur, IN

Abstract

In what follows, P_m(n) denotes the number of those partitions of n in which the smallest element is m; p_m(n) the number of those in which m is the largest element, i.e. the number of partitions of n into exactly m non-zero parts; and p(n) denotes the number of unrestricted partitions of n.

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Hansraj Gupta ¹

Affiliations
1 Government College, Hoshiarpur, IN

Abstract

Throughout this paper, p denotes an odd prime; all other small letters except x, denote integers ≥0, unless stated otherwise. Φ is Euler's function. The symbol " < " is read "less than and prime to."

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Hansraj Gupta ¹

Affiliations
1 Hoshiarpur, IN

Abstract

Let k(n) denote the number of integers not exceeding n, which cannot be expressed as the sums of less than four squares. Then since all such numbers are of the form 4^a(8b+7), where a and b are non-negative integers.

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