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REVIEW 3 major objections 4 minor 38 references

On Some New Congruences For Biregular Overpartitions

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper establishes new infinite families of congruences for biregular overpartition counts modulo 4, 8, 6, and 12 along arithmetic progressions governed by powers of 2 and 3.

desk verdict New infinite family congruences for biregular overpartitions, with proof gaps that are real but likely repairable. read the letter →

arxiv 2507.02720 v1 pith:DDPQETX2 submitted 2025-07-03 math.NT

Anakha V This is my paper
classification math.NT MSC 11P8305A17
keywords biregularoverpartitionscongruencesq-seriesdissectionformulasthetafunctionsarithmeticprogressionsRamanujan-type
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to prove infinite families of congruences for the biregular overpartition function $B_{\ell_1,\ell_2}(n)$, which counts overpartitions of $n$ whose parts are divisible by neither $\ell_1$ nor $\ell_2$. Taking $\ell_1$ to be a power of 2 and $\ell_2$ a power of 3, it shows that on certain arithmetic progressions the count is always divisible by 4, 8, 6, or 12. For instance, it claims $B_{2^\alpha,3}(4n+2)\equiv0\pmod4$ for $\alpha\ge2$, $B_{2^\alpha,3}(8n+6)\equiv0\pmod8$ for $\alpha\ge3$, and corresponding families where the prime powers in the excluded moduli vary independently. If correct, these results generalize earlier special cases from [26] and [15] to arbitrary powers of 2 and 3 using only generating functions, dissection formulas, and $\theta$ functions. A reader should care because the congruences are new infinite families, not isolated examples, and the method is elementary rather than algorithmic.

What carries the argument

The load-bearing object is the generating function (1.1) for $B_{\ell_1,\ell_2}(n)$, an infinite product built from $(q;q)_\infty$ factors. The argument proceeds by q-series dissection: known dissections, such as the 2-dissections (2.2)--(2.5) and the 3-dissections (2.6)--(2.7), split the product into residue classes, and extracting coefficients in a fixed class leaves a prefactor that visibly divides 4, 8, 12, or 6. The $\theta$ series $\varphi(-q)=(q;q)_\infty^2/(q^2;q^2)_\infty$ encodes the square-of-squares structure used to locate the vanishing residue classes. For the modulo-3 part, the congruence $(q;q)_\infty^p\equiv(q^p;q^p)_\infty\pmod p$ (stated as Lemma 2.1) is used to replace factors with $q^3$ by factors in $q$.

What would settle it

Expand the generating function (1.1) with a computer algebra system for a small case, say $\ell_1=8$, $\ell_2=3$, and check whether $B_{8,3}(8n+6)\equiv0\pmod8$ for $n=0,1,\dots,20$; any violation refutes Theorem 1.1. For Theorem 1.2, verify the coefficient congruence claimed in (3.10)--(3.11) modulo 3 on the first few powers of $q$ to see whether Lemma 2.1's substitution actually holds.

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Extended reading notes

Core claim

The central claim is that the biregular overpartition count obeys infinitely many congruence identities indexed by the exponents of 2 and 3 in the excluded moduli. For every $n\ge0$ and $\alpha\ge2$, $B_{2^\alpha,3}(4n+2)\equiv0\pmod4$ and $B_{2^\alpha,3}(8n+5)\equiv0\pmod4$, while for $\alpha\ge3$ $B_{2^\alpha,3}(8n+6)\equiv0\pmod8$. Theorem 1.2 asserts $B_{2^\alpha,3}(4^k(4n+3))\equiv0\pmod6$ when $\alpha>2k+1$ and $B_{2^\alpha,3}(8n+7)\equiv0\pmod{12}$ for $\alpha>2$. Theorem 1.3 gives mod-4 and mod-8 families for $B_{2^{2\alpha+1},3^\beta}(9n+3i)$ and $B_{2^{2\alpha},3^\beta}(9n+3i)$ with $i=1,2$, and Theorem 1.4 gives mod-8 results at $12n+3$, $12n+7$, and $12n+11$ depending on the parity of $\alpha$ and $\beta$. Taken together, the paper claims the full set of arithmetic progressions listed in Theorems 1.1 through 1.4.

Load-bearing premise

The proof of Theorem 1.2 leans on Lemma 2.1, a congruence stated with an undefined symbol and no proof; the step actually used is the replacement $(q^3;q^3)_\infty^2\equiv(q;q)_\infty^6\pmod3$ inside equation (3.10). If that substitution is not valid in the form applied, the modulo-3 argument behind Theorem 1.2 fails.

Editorial extensions

If this is right

  • The congruences give infinite arithmetic progressions on which the biregular overpartition counts are divisible by 4, 8, 12, or 6, so the vanishing is not a finite coincidence.
  • The results extend known congruences for the pairs $(4,3)$, $(4,9)$, $(8,3)$, and $(8,9)$ treated in [26] and [15] to all exponents $\alpha\ge2$ and $\beta\ge2$ where the theorems apply.
  • The elementary dissection method offers a template for proving similar congruences for other pairs $(\ell_1,\ell_2)$ without invoking a general algorithm.
  • The mod-8 and mod-12 statements are stronger than mere parity, refining the known fact that $B_{\ell_1,\ell_2}(n)\equiv0\pmod2$ for $n\ge1$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If Lemma 2.1 is repaired, the same reduction to $(q;q)_\infty^p\equiv(q^p;q^p)_\infty\pmod p$ could likely push the mod-3 and mod-6 congruences of Theorem 1.2 to exponents outside the stated range $\alpha>2k+1$.
  • The $q^{4n+3}$-vanishing argument resembles the classical fact that a sum of two squares cannot be $3\pmod4$; a natural test is whether similar square-of-squares obstructions produce mod-8 congruences for other excluded moduli.
  • One can numerically probe whether the theorem's exponent thresholds are sharp: for example, check whether $B_{2,3}(4n+2)\equiv0\pmod4$ fails, which would justify the $\alpha\ge2$ hypothesis in Theorem 1.1.
  • A natural next step would be to test whether the same generating function yields mod-9 or mod-27 congruences for these families; the paper does not pursue them.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the generating function for (ℓ1,ℓ2)-biregular overpartitions B_{ℓ1,ℓ2}(n) and proves four families of congruences. Theorem 1.1 gives congruences modulo 4 and 8 for B_{2^α,3}(n) in the progressions 4n+2, 8n+5, and 8n+6. Theorem 1.2 gives a modulo 6 family for B_{2^α,3}(4^k(4n+3)) and a modulo 12 congruence for B_{2^α,3}(8n+7). Theorem 1.3 gives modulo 4 and 8 congruences for B_{2^{2α+1},3^β}(9n+3i) and B_{2^{2α},3^β}(9n+3i). Theorem 1.4 gives parity-dependent modulo 8 congruences for B_{2^α,3^β}(12n+3), (12n+7), and (12n+11). The methods are elementary q-series manipulations using 2- and 3-dissections and congruence substitutions. The main theorems generalize results from the recent papers [26] and [15] from fixed small pairs to arbitrary powers of 2 and 3.

Significance. If the stated congruences are correct, the paper provides genuinely new infinite families of congruences for biregular overpartitions and extends several recent results in a uniform way. The proofs are elementary and mostly checkable; there is no circular reasoning, since the target congruences are not used as assumptions. The explicit computations in equations (3.5)-(3.7) and (4.4) are largely verifiable, which is a strength. The manuscript does not include machine-checked proofs or code, so its value depends on the clarity and completeness of the q-series manipulations. A few load-bearing passages are currently sketched or rest on an ill-stated lemma; these need to be repaired before the results can be certified.

major comments (3)
  1. [Section 2, Lemma 2.1; Section 3, (3.10)-(3.11)] Lemma 2.1 as printed is not a meaningful statement: the symbol f is never defined, and 'f_{p^{k-1}} p^m ≡ f_{p^k} m (mod p^k)' is not a congruence between q-series in any standard notation. The proof of Theorem 1.2 then invokes (2.1) to pass from (3.1) to (3.10)-(3.11), replacing (q^3;q^3)_∞^2 by (q;q)_∞^6, (q^6;q^6)_∞ by (q^2;q^2)_∞^3, (q^{2^{α+1}3};...)_∞ by (q^{2^{α+1}};...)_∞^3, and (q^{2^α3};...)_∞^2 by (q^{2^α};...)_∞^6 modulo 3. The standard congruence the proof needs is (q;q)_∞^p ≡ (q^p;q^p)_∞ (mod p), which is not what Lemma 2.1 states and is not proved or referenced. Since this reduction is the only route to the modulo-3 part of Theorem 1.2, the proof as written is incomplete at a load-bearing point. The lemma should be restated correctly, proved or cited, and the substitutions in (3.10)-(3.11) should be written out.
  2. [Section 3, (3.25) and (3.28)] In the proof of Theorem 1.3, after deriving the 3-dissections (3.25) and (3.28), the paper simply says 'From this we can see that (1.7) holds' and 'Accordingly, (3.28) implies (1.8)'. To justify the theorem one must show that the right-hand side of (3.25) has vanishing coefficients in degrees 3n+1 and 3n+2 modulo 4, and similarly for (3.28) modulo 8. This does follow, because β≥2 makes every base appearing in (3.25) a multiple of 3 and the denominator is a unit, but the coefficient extraction is not demonstrated. Please include the extraction step.
  3. [Section 3, proof of Theorem 1.1, after (3.8)] The proof of congruence (1.3) is omitted. The text says that after gathering even powers from (3.8), combining with (2.2), (2.4), and (2.5), and then collecting q^{2n+1} terms, 'one can obtain (1.3)'. No intermediate identity is displayed. Because (1.3) is one of the three assertions of Theorem 1.1, the derivation should be provided, at least in the form of the resulting generating function for Σ B_{2^α,3}(8n+5)q^n with its explicit factor of 4.
minor comments (4)
  1. [Throughout] Many q-series displays are missing or misplacing parentheses; for example, equations (3.3), (3.4), and (3.5) contain fragments like 'q2α·3; q2α·3)∞' instead of '(q^{2^α·3}; q^{2^α·3})∞'. This makes the computations difficult to read and should be corrected throughout.
  2. [Abstract and Introduction] The abstract uses \bar{B}_{ℓ1,ℓ2}(n) while the body uses B_{ℓ1,ℓ2}(n); please use one notation consistently.
  3. [Section 4, paragraph before (4.4)] The text says 'For α ≥ 2, φ(−q^{2^α·3^α}) contains only the terms of the form q^{12k}'; the exponent 3^α should presumably be 3^β to match the denominator in (4.3).
  4. [Section 4, (4.3)] The identity (φ(q^i))^j ≡ 1 (mod 8) for j ≥ 4 a multiple of 4 is used without proof; a one-line binomial justification would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the congruences are derived from the stated generating function, external dissection lemmas, and an external parity result. The malformed Lemma 2.1 is a proof gap, not a circular step.

full rationale

The paper's derivation chain starts from the generating function (1.1), which follows directly from the definition of (l1,l2)-biregular overpartitions, and then applies standard external dissection formulas (Lemmas 2.2-2.4) and cited parity results. The target congruences, e.g. B_{2^alpha,3}(4n+2) ≡ 0 mod 4, are never assumed at the start of any proof; they are obtained at the end of each computation after extracting specific q-powers. The only noteworthy defect is Lemma 2.1, whose printed statement is malformed: the symbol f is undefined, and the congruence 'f_{p^{k-1}} p^m ≡ f_{p^k} m (mod p^k)' is not a meaningful q-series identity as written. In the proof of Theorem 1.2 the author uses what is evidently the classical congruence (q;q)_∞^p ≡ (q^p;q^p)_∞ mod p, replacing (q^3;q^3)_∞^2 by (q;q)_∞^6 and (q^6;q^6)_∞ by (q^2;q^2)_∞^3 modulo 3. This standard identity, if supplied with a correct statement and proof, would support the reduction, and it is not a restatement of the paper's conclusions. Thus the issue is an unproved and inaccurately stated lemma, not circular reasoning. There is no self-citation chain carrying the argument, no fitted parameter renamed as a prediction, and no theorem being imported from the present author's prior work. The paper is accordingly assigned a circularity score of 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proofs rely on standard q-series dissection identities and one cited prior congruence. There are no free parameters fitted to data and no new postulated entities. The ambiguity in Lemma 2.1 is a notational issue but not a circular one.

assumptions (4)
  • standard math Lemma 2.1: congruence (q;q)_infty^p == (q^p;q^p)_infty (mod p) for prime p, used in the proof of Theorem 1.2.
    The lemma is stated ambiguously with undefined notation f, but the intended identity is a standard Freshman's dream congruence for q-products. It is invoked in Section 3 to derive (3.10), which is load-bearing for the modulo 3 part of Theorem 1.2.
  • standard math Dissection formulas (2.2), (2.3), (2.4), (2.5), (2.6), (2.7) from the prior literature.
    These identities are cited from Xia-Yao, Barua-Ojah, Ramanujan, and Hirschhorn-Sellers. The proofs of all theorems depend on them, and the paper does not prove them.
  • standard math Identity 1/phi(-q) = phi(q) phi(q^2)^2 phi(q^4)^4 ..., with (phi(q^i))^j == 1 (mod 8) for j >= 4 a multiple of 4.
    Used in the proof of Theorem 1.4 to reduce the generating function modulo 8. This is a known identity, but the paper states it without proof.
  • domain assumption Prior result from [3]: B_{2^alpha,3}(n) == 0 (mod 2) for n >= 1.
    Used in Theorem 1.2 to combine modulo 2 and modulo 3 congruences into modulo 6. This is an external theorem the paper relies on.

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Cite this review

Pith. "Pith review of On Some New Congruences For Biregular Overpartitions." pith.science (2026). https://pith.science/paper/DDPQETX2

@misc{pith2026250702720,
  author       = {Pith},
  title        = {Pith review of: On Some New Congruences For Biregular Overpartitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DDPQETX2}},
  note         = {Machine review of arXiv:2507.02720}
}
abstract

Inspired by the recent work by Nadji, Ahmia and Ram\'irez, we examined the arithmetic properties of $\bar{B}_{l_1,l_2} (n)$, the number of overpartitions of n whose parts are neither divisible by $l_1$ nor divisible by $l_2$. In particular, we establish some congruences modulo k in {4, 8, 6, 12} satisfied by $\bar{B}_{l_1,l_2} (n)$ where $l_1$ and $l_2$ take values as arbitrary powers of 2 and 3. Moreover, we extend certain results proved in [26] and [15] for $l_1$ and $l_2$ with random powers of 2 and 3. Generating functions, dissection formulas, and theta functions are used to prove our main findings.

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Reference graph

Works this paper leans on

38 extracted references · 37 canonical work pages

  1. [26]

    Nadji, Moussa Ahmia, and Jos ´e L

    Mohammed L. Nadji, Moussa Ahmia, and Jos ´e L. Ram´ırez, Arithmetic properties of biregular overpartitions, The Ramanujan Journal, 67(1):1–21, 2025

    work page 2025

  2. [15]

    Suparno Ghoshal and Arijit Jana, Further results on arithmetic properties of biregular overpartitions , arXiv preprint arXiv:2504.21439, 2025

    arXiv 2025

  3. [1]

    Some new congruences for (s, t)-regular bipartition func- tions

    Chandrashekar Adiga, Abdelmejid Bayad, and Ranganatha Dasappa. Some new congruences for (s, t)-regular bipartition func- tions. Publications de l’Institut Mathematique, 106(120):53–67, 2019

    work page 2019

  4. [2]

    Zakir Ahmed and Nayandeep Deka Baruah, New congruences for Andrews’ singular overpartitions , International Journal of Number Theory, 11(07):2247–2264, 2015

    work page 2015

  5. [3]

    Some Properties of Overpartitions into Nonmultiples of Two Integers

    Abdulaziz M. Alanazi, Augustine O. Munagi, and Manjil P. Saikia, Some properties of overpartitions into nonmultiples of two integers, arXiv preprint arXiv:2412.18938, 2024

    work page Pith review arXiv 2024

  6. [4]

    Alanazi, Augustine O

    Abdulaziz M. Alanazi, Augustine O. Munagi, and James A. Sellers, An infinite family of congruences for ℓ-regular overparti- tions, Integers, 16:A37, 2016

    work page 2016

  7. [5]

    Andrews, Singular overpartitions, International Journal of Number Theory, 11(5):1523–1533, 2015

    George E. Andrews, Singular overpartitions, International Journal of Number Theory, 11(5):1523–1533, 2015

    work page 2015

  8. [6]

    Rupam Barman and Chiranjit Ray, Divisibility of Andrews’ singular overpartitions by powers of 2 and 3 , Research in Number Theory, 5:1–7, 2019

    work page 2019

Show all 38 references
  1. [7]

    1–17, 2025

    Nayandeep Deka Baruah and Subhajit Bandyopadhyay, On some restricted overpartition functions, Indian Journal of Pure and Applied Mathematics, pp. 1–17, 2025

    2025

  2. [8]

    Nayandeep Deka Baruah and Kanan Kumari Ojah, Analogues of Ramanujan’s partition identities and congruences arising from his theta functions and modular equations, The Ramanujan Journal, 28(3):385–407, 2012

    2012

  3. [9]

    Berndt, Ramanujan’s Notebooks: Part III, Springer, Germany, 1991

    Bruce C. Berndt, Ramanujan’s Notebooks: Part III, Springer, Germany, 1991

    1991

  4. [10]

    P3–17, 2015

    Kathrin Bringmann, Jehanne Dousse, Jeremy Lovejoy, and Karl Mahlburg, Overpartitions with restricted odd differences, The Electronic Journal of Combinatorics, pp. P3–17, 2015

    2015

  5. [11]

    Hirschhorn, and James A

    Shi-Chao Chen, Michael D. Hirschhorn, and James A. Sellers, Arithmetic properties of Andrews’ singular overpartitions, Inter- national Journal of Number Theory, 11(05):1463–1476, 2015

    2015

  6. [12]

    Sylvie Corteel and Jeremy Lovejoy, Overpartitions, Transactions of the American Mathematical Society, 356(4):1623–1635, 2004

    2004

  7. [13]

    Gu, Arithmetic properties of ℓ-regular partitions , Advances in Applied Mathematics, 51(4):507–523, 2013

    Su-Ping Cui and Nancy S.S. Gu, Arithmetic properties of ℓ-regular partitions , Advances in Applied Mathematics, 51(4):507–523, 2013

    2013

  8. [14]

    Dou, Congruences for (3, 11)-regular bipartitions modulo 11, The Ramanujan Journal, 40:535–540, 2016

    Donna Q.J. Dou, Congruences for (3, 11)-regular bipartitions modulo 11, The Ramanujan Journal, 40:535–540, 2016

    2016

  9. [16]

    Hao and Erin Y .Y

    Robert X.J. Hao and Erin Y .Y . Shen, On the number of ℓ-regular overpartitions, International Journal of Number Theory, 17(09):2153–2173, 2021

    2021

  10. [17]

    Hirschhorn, The power of q, Developments in Mathematics, 49, 2017

    Michael D. Hirschhorn, The power of q, Developments in Mathematics, 49, 2017

    2017

  11. [18]

    Hirschhorn and James A

    Michael D. Hirschhorn and James A. Sellers, An infinite family of overpartition congruences modulo 12 , Integers, 5(1):A20, 2005

    2005

  12. [19]

    Hirschhorn and James A

    Michael D. Hirschhorn and James A. Sellers, Elementary proofs of parity results for 5-regular partitions, Bulletin of the Aus- tralian Mathematical Society, 81(1):58–63, 2010

    2010

  13. [20]

    Hirschhorn and James A

    Michael D. Hirschhorn and James A. Sellers, A congruence modulo 3 for partitions into distinct non-multiples of four , Journal of Integer Sequences, 17(9):14–9, 2014

    2014

  14. [21]

    Hirschhorn and James A

    Michael D. Hirschhorn and James A. Sellers, Congruences for overpartitions with restricted odd differences , The Ramanujan Journal, 53(1):167–180, 2020. 12 ANAKHA V

    2020

  15. [22]

    Byungchan Kim, The overpartition function modulo 128, Integers, 8(1):i38–pdf, 2008

    2008

  16. [23]

    Byungchan Kim, A short note on the overpartition function, Discrete Mathematics, 309(8):2528–2532, 2009

    2009

  17. [24]

    Lin, Arithmetic of the 7-regular bipartition function modulo 3, The Ramanujan Journal, 37(3):469–478, 2015

    Bernard L.S. Lin, Arithmetic of the 7-regular bipartition function modulo 3, The Ramanujan Journal, 37(3):469–478, 2015

    2015

  18. [25]

    Jeremy Lovejoy, Gordon’s theorem for overpartitions, Journal of Combinatorial Theory, Series A, 103(2):393–401, 2003

    2003

  19. [27]

    Sellers, and Haiyang Wang, Extending recent congruence results on (ℓ, µ)-regular overpartitions, arXiv preprint arXiv:2505.21989, 2025

    Bishnu Paudel, James A. Sellers, and Haiyang Wang, Extending recent congruence results on (ℓ, µ)-regular overpartitions, arXiv preprint arXiv:2505.21989, 2025

    2025 arXiv

  20. [28]

    Cristian-Silviu Radu, An algorithmic approach to Ramanujan–Kolberg identities , Journal of Symbolic Computation, 68:225–253, 2015

    2015

  21. [29]

    Chiranjit Ray, Distribution of Andrews’ singular overpartitions ¯cp,1(n), Research in Number Theory, 10(1):10, 2024

    2024

  22. [30]

    Nipen Saikia and Chayanika Boruah, Congruences for ℓ-regular overpartition for {5, 6, 8}, Indian Journal of Pure and Applied Mathematics, 48:295–308, 2017

    2017

  23. [31]

    Shen,Arithmetic properties of ℓ-regular overpartitions, International Journal of Number Theory, 12(03):841–852, 2016

    Erin Y .Y . Shen,Arithmetic properties of ℓ-regular overpartitions, International Journal of Number Theory, 12(03):841–852, 2016

    2016

  24. [32]

    Ajit Singh and Rupam Barman, New density results and congruences for Andrews’ singular overpartitions, Journal of Number Theory, 229:328–341, 2021

    2021

  25. [33]

    Nicolas Allen Smoot, On the computation of identities relating partition numbers in arithmetic progressions with eta quotients: an implementation of Radu’s algorithm, Journal of Symbolic Computation, 104:276–311, 2021

    2021

  26. [34]

    Nabin Kumar Meher and Sourav Bhowmick, Some new congruences on biregular overpartitions , arXiv preprint arXiv:2507.01529, 2025

    2025 arXiv

  27. [35]

    Liuquan Wang, Arithmetic properties of ℓ-regular bipartitions, Bulletin of the Australian Mathematical Society,95(3):353–364, 2017

    2017

  28. [36]

    Xia, Congruences modulo 9 and 27 for overpartitions, The Ramanujan Journal, 42:301–323, 2017

    Ernest X.W. Xia, Congruences modulo 9 and 27 for overpartitions, The Ramanujan Journal, 42:301–323, 2017

    2017

  29. [37]

    Xia and Olivia X.M

    Ernest X.W. Xia and Olivia X.M. Yao, Analogues of Ramanujan’s partition identities, The Ramanujan Journal, 31(3):373–396, 2013

    2013

  30. [38]

    Yao and Ernest X.W

    Olivia X.M. Yao and Ernest X.W. Xia, New Ramanujan-like congruences modulo powers of 2 and 3 for overpartitions, Journal of Number Theory, 133(6):1932–1949, 2013. ANAKHA V, NATIONAL INSTITUTE OF TECHNOLOGY CALICUT, KOZHIKODE, KERALA, INDIA 673601 Email address: anakha p240019m...

    1932

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