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# A partition of n is a non-increasing sequence of positive integers, called parts, whose sum is n. A partition of n is called a t-core of n if none of the hook numbers is a multiple of t. If a

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## Congruences modulo 3 for two interesting partitions arising from two theta function identities

### Kuwali Das

Department of Mathamatical Sciences, Bodoland University, Kokrajhar-783370, Assam, INDIA kwldas90@gmail.com

Abstract. We find several interesting congruences modulo 3 for 5-core partitions and two color partitions.

Keywords:t-core partition, Theta function, Dissection, Congruence.

MSC 2000 classification:MSC 2000 classification: primary 11P83; secondary 05A17

t

t

n=0

t

n

t

t

t

n=1

n

5

### (n) in terms of the prime factorization of n + 1 can be found in [5, Theorem 4]. This theorem follows from an identity of Ramanujan recorded in his famous manuscript on the partition function and tau-function now published with his lost notebook [6, p. 139].

http://siba-ese.unisalento.it/ c 2016 Universit`a del Salento

(2)

c

a11

ass

b11

att

i

j

5

c

s i=1

aii+1

i

t j=1

jbj+1

j

5

α

α

5

α

5

α

α+1

α

5

α+1

α

k

k

n=0

k

n

k

k

k

5

t−1 k=0

k

k

t

k

t

(3)

n=−∞

n(n+1)/2

n(n−1)/2

2

2

2

2

2

2

2

2

5

2

4

4

2

3

2

2

2

2

2

2

2

2

4

4

2

2

2

2

2

2

4

4

2

2

(4)

p−1

2

k=p−12 k6=±p−16

k

3k2+k2

3p2+(6k+1)p

2

3p2−(6k+1)p 2

±p−16

p2−124

p2

p2

2

2

2

2

2

10

10

2

2

6

6

2

30

30

2

6

6

4

4

30

30

4

2

2

5

5

20

2

5

2

5

2

2

2

5

4

4

5

2

5

5

20

2

2

5

4

4

5

2

2

2

10

10

2

(5)

2

2

2

10

10

2

3

5

15

9

3

15

5

5

45

5

15

75

2

2

2

10

10

2

3

9

3

3

15

15

45

5

15

15

75

5

2

2

7

3

3

12

12

3

4

4

3

6

6

3

3

3

5

2

2

4

2

2

2

10

10

2

3

6

6

2

30

30

2

6

6

4

5

30

30

4

QED

n=0

5

n

3

3

5

5

n=0

5

n

15

15

n=0

5

n

n=0

5

n

(6)

2

n=0

5

2n

10

10

5

2

2

n=0

5

2n

30

30

2

2

2

10

10

2

6

6

n=0

5

n

30

30

6

6

2

6

6

2

30

30

2

6

6

4

4

30

30

4

2

6

6

30

30

3

6

6

3

30

30

4

30

30

5

6

6

2

6

6

90

90

18

18

30

30

4

n=0

5

6n

6n

6n+2

6n+4

QED

5

k

k

k

5

5

5

5

5

(7)

25

25

5

5

2

5

5

10

15

5

20

5

20

10

15

3

n=0

5

n

5

5

75

75

15

15

3

6

15

15

5n+3

n=0

5

n

15

15

5n+2

5n+4

QED

5

5

5

5

n=0

5

n

15

15

25

25

5

5

2

5

5

5n+1

(8)

n=0

5

n

3

3

5

5

5n+3

5n+4

QED

5

2k

2k

5

n=0

5

n

p−1

2

k=−p−12 k6=±p−16

k

3·3k2+k2

3·3p2+(6k+1)p

2

3·3p2−(6k+1)p

2

±p−16

p2−124

3p2

p−1

2

k=p−12 k6=±p−16

k

5·3k2+k2

5·3p2+(6k+1)p

2

5·3p2−(6k+1)p

2

±p−16

5·p2−124

5p2

2

2

2

2

2

(9)

2

2

2

pn+8·p2−124

n=0

5

2

n

n=0

5

2

n

3p

3p

5p

5p

n=0

5

2

2

n

3

3

5

5

5

2

2

5

QED

5

2k+2

2k+1

n=0

5

2k

2

2k

n

3p

3p

5p

5p

n=0

5

2k+1

2k+2

n

3p

3p

5p

5p

5

2k+1

2k+2

QED

(10)

5

2k

2k

n=0

5

2k

2k

n

3

3

5

5

3

3

3

5

5

5n+r

5

2k

2k

QED

5

2k

2k

5

n=0

5

n

p−1

2

k=p−12 k6=±p−16

k

3k2+k2

3p2+(6k+1)p

2

3p2−(6k+1)p

2

±p−16

p2−124

p2

p−1

2

k=p−12 k6=±p−16

k

15·3k2+k2

15·3p2+(6k+1)p

2

15·3p2−(6k+1)p

2

±p−16

15·p2−124

15p2

2

2

2

(11)

2

2

2

2

2

pn+16·p2−124

n=0

5

2

n

n=0

5

2

n

p

p

15p

15p

n=0

5

2

2

n

15

15

5

2

2

5

QED

5

2k+2

2k+1

5

2k

2k

(12)

5

n=0

5

n

3

3

5

5

n=0

5

n

5

5

n=0

5

n

15

15

n=0

5

n

5

5

n=0

5

n

5

5

5

5

5

5

5

6

n=0

5

n

5

5

5

15

15

2

15

15

2

n=0

5

n

3n

3n+1

3n+2

n=0

5

n

5

5

2

n=0

5

n

n=0

5

n

5

5

2

n=0

5

n

(13)

n=0

5

n

5

5

2

n=0

5

n

QED

5

5

5

5

n

3

n=0

5

n

75

75

5

5

15

15

3

6

15

15

5n+3

5n+2

5n+4

QED

n=0

5

2k−1

2k−1

n

k+1

5

5

n=0

5

n

5

5

n=0

5

2k−1

2k−1

n

k+1

3

3

15

15

2

5

5

2

k+1

3

3

15

15

n=0

5

n

2

(14)

3n+2

n=0

5

2k−1

2k−1

n

k+1

5

5

n=0

5

n

2

n=0

5

n

n=0

5

n

n=0

5

2k−1

2k−1

n

k+1

5

5

2

5

5

2

3

3

15

15

5

5

k+2

5

5

QED

5

5

5n+r

n=0

5

2k−1

2k−1

n

5

2k

2k

5

(15)

n=0

5

n

p−1

2

k=p−12 k6=±p−16

k

3k2+k2

3p2+(6k+1)p

2

3p2−(6k+1)p

2

±p−16

p2−124

p2

p−1

2

k=p−12 k6=±p−16

k

3k2+k2

3p2+(6k+1)p

2

3p2−(6k+1)p

2

±p−16

5·p2−124

5p2

2

2

2

2

2

2

2

2

pn+·p2−14

n=0

5

2

n

n=0

5

2

n

p

p

5p

5p

n=0

5

2

2

n

5

5

5

2

2

5

(16)

QED

5

2k+2

2k+1

5

2k

2k

### References

[1] Z. Ahmed, N. D. Baruah and M. G. Dastidar:New congruences modulo 5 for the number of 2-color partitions, J. Number Theory, 157 (2015), 184–198.

[2] N. D. Baruah, J. Bora and K. K. Ojah: Ramanujan’s modular equations of degree 5, Proc. Indian Acad. Sci. (Math. Sci.), 122 (2012), 485–506.

[3] B. C. Berndt: Ramanujan’s Notebooks, Part III, Springer-verleg, New York, 1991.

[4] S. P. Cui and N. S. S. Gu: Arithmetic properties of ℓ-regular partitions, Adv. Appl.

Math., 51 (2013),507–523

[5] F. Garvan, D. Kim and D. Stanton: Cranks and t-cores, Invent. Math., 101(1990), 1–17.

[6] S. Ramanujan: The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.

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