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Computational Search for Gaussian Perfect IntegersAshmi Banerjee*, Shaunak Mukherjee*, Somjit Datta**, Subhashis

Majumder*

*Department of Computer Science and Engineering,**Department of Mathematics,

Heritage Institute of Technology, Kolkata

SOME BASIC QUESTIONS.

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• What is a Gaussian Integer?

• What is a Gaussian Prime?

• What is a Norm?

• What is a Gaussian Primary Prime?

• What is a Perfect Number?

• Gaussian Integer A complex number on the Argand plane, referred to as a + bi where a,b є Z.

EXAMPLE:2 + 3i, -5 - 9i are Gaussian Integers.2.5 + 5.6i is NOT a Gaussian Integer.

• Associates of a Gaussian Integer

A Gaussian Integer with a particular norm has four associates: a + bi -a - bi -b + ai b - ai.

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• What is a Norm?



• Gaussian Primes

We can characterize Gaussian primes as follows:

• If (a, b) ≠0, then a + bi is a Gaussian prime ifa2+b2 = p, where p is a prime in the rational domain.

• A Gaussian integer of the form of a or ai, a є Z, isa Gaussian prime if a is a rational prime and |a|≡ 3(mod 4).

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• What is a Norm?



• Norm:

Norm = sum of the squares of the real andimaginary portions of the Gaussian integer.

EXAMPLE:For the Gaussian Integer 2+3i, the norm is 2*2+3*3=13

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• What is a Norm?



• Gaussian primary prime

The number x + yi out of those 4 characterizedassociates can be called a Gaussian Primary Prime when simultaneously either

x ≡ 1; y ≡ 0 (mod 4)ORx ≡ -1; y ≡ 2 (mod 4)

• EXAMPLES OF GAUSSIAN PRIMARY PRIMES UPTO A NORM OF 200

Primary Gaussian Primes Norm

1+1i 2

-1+2i 5

-1-2i 5

1+4i 17

1-4i 17

-1+6i 37

-1-6i 37

-1+10i 101

-1-10i 101

-1+14i 197

-1-14i 197

OUR FIRST RESULT OBTAINED

A list of Gaussian primes and Gaussian primary primes up-to a norm of

10,00,000.

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• What is a Norm?



• Perfect Number

A perfect number in the rational domain, is a positive integer that is equal to the sum of its proper positive divisors.

EXAMPLE:

6 has the proper divisors as 1,2,3.So, the sum of the proper divisors=1+2+3=6, which is the number itself.

Hence, 6 is a Perfect Number in the Integer domain.

• Factorization of the first 3 Perfect Integers:

Number Proper Divisors Sum of Proper Divisors

6 1,2,3 6

28 1,2,4,7,14 28

496 1,2,4,8,16,31,62,124,248

496

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What is a Gaussian Perfect Integer? Does it really exist?

And, that is the problem.

Gaussian Perfect Integer

A Gaussian perfect integer is one which is equal to the sum of its proper divisors

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Two Interpretations of the Problem Definition

• The Factorization Technique

• The Norm Perfect Technique

I. FACTORIZATION TECHNIQUE

Norm Perfect Technique 21

Basic Algorithm:

INPUT: A Gaussian Integer Z= a + bi

OUTPUT: YES if Z is a Perfect Gaussian Integer ; NO if it isn’t.

Detailed Algorithm

STEP 1. Calculate modulus of the Gaussian Integer.STEP 2. Prime Factorize the modulus and express it

as a combination of prime factors.STEP 3. Generate all the factors of the modulus as a

product of the various combination of prime factors obtained in STEP 2.

STEP 4. Replace every prime factor with its equivalent Gaussian primary prime from the list obtained in Result 1.

Detailed Algorithm continued…

STEP 5. Compute the product of the Gaussian primary primes for each factor of the modulus using the product function in <complex.h>

STEP 6. Compute the sum of the real and imaginary components of the factors and check if it is equal to the initial Gaussian Integer.

If YES, then it is a Perfect Gaussian Integer, and if NO, it obviously isn’t.

Factorization Technique 24

An example:

34 + 24i Modulus (Norm) : 1732 The sum of all the proper divisors = (1 + i) + (0 + 2i) + (17 − 12i) + (29 + 5i)= 47 − 4i

BUT, 47-4i != 34+24i

So, 34+24i is not a Gaussian Perfect Integer as per the Factorization technique.

EXPERIMENTAL RESULTS

RESULT OF FACTORIZATION TECHNIQUE

NO Gaussian Perfect Integers found within a norm of

1,00,000.


Hardware Specifications

• Intel i5 Personal Computer • 3GB RAM• Programs have been written in C• CPU processor speed = 2.5 GHz• Tested successfully on both Windows 7 and

Fedora Linux platforms.


Computational Time Required

Norm Limit Time needed (seconds)

41,210 35.518

~50,000 ~60

1,00,000 ~600

Graphical extrapolation of moduli differences through Factorization method :


•Time required for each Gaussian integer Z, is dominated by the factorization of the norm of Z (say n)•This will need square-root (n) time

Entire algorithm complexity : O(n3/2)

II. NORM-PERFECT TECHNIQUE


Basic Algorithm:

INPUT: A Gaussian Integer Z= a + bi

OUTPUT: YES if Z is a Perfect Gaussian Integer ; NO if it isn’t.

Detailed AlgorithmSTEP 1. Calculate modulus of the Integer.STEP 2. Use division function in <complex.h> header file to implement

normal factorization technique.STEP 3. Isolate proper Gaussian divisors of the Integer through STEP 2.STEP 4. Calculate moduli of each proper divisor, and continue adding

them.STEP 5. Check if the summation of the moduli equals the modulus of

the initial Integer. If YES, then it is a Perfect Gaussian Integer, and if NO, it obviously

isn’t.


An example:34 + 24iModulus = 1732Proper divisors (using the division function in <complex.h>) : 0+2i, 1+1i, 2+0i, 17+12i, 29+5iTheir moduli : 4, 2, 4, 433, 866Summation = 1309 ≠ 1732

Hence, this is definitely not a norm-perfect Gaussian integer as per the definition.

EXPERIMENTAL RESULTS

RESULT OF NORM PERFECT TECHNIQUE

NO Gaussian Perfect Integers found within a norm of

10,00,000.


Hardware Specifications

• Intel i5 Personal Computer • 3GB RAM• Programs have been written in C• CPU processor speed = 2.5 GHz• Tested successfully on both Windows 7 and

Fedora Linux platforms.


Computational Time Required

Norm Limit Time needed (seconds)

200 2.787

20,000 16.031

1,00,000 49.527

Graphical Extrapolation using differences in moduli as per Norm-Perfect Technique (1)

Graphical Extrapolation using differences in moduli as per Norm-Perfect Technique (2)

Norm Perfect Technique

• No. of computations required to prime-factorize the Integer : linearly related to number of Primes required to divide the Integer• Straight-up approach followed in dividing with all Gaussian Integers whose norm < square-root (norm of required Integer)• No. of proper divisors will depend on density of Integers on Argand plane

Hence, the complexity calculation of the technique is a non-trivial process.

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CONCLUSION AND FUTURE WORK

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• Sums of moduli of norms of proper divisors (Norm-Perfect technique) are not equal to norms of actual Gaussian integers.

• Sums of proper divisors (Factorization technique) are not equal to the corresponding Gaussian integers.

SO.Non-existence of any Gaussian perfect integers, up to a

very large norm, is thus computationally demonstrated.

N.B. Bound can be improved.

References[1] Richard P. Brent and Graeme L. Cohen: A New LowerBound for Perfect Integers, Mathematics of Computation,Volume 53, No. 187, pp 431-437.

[2] G. Dandapat, J. L. Hunscuker and Carl Pomerance: SomeNew Results on Odd Perfect Numbers, Pacific Journal ofMathematics, 1975.

[3] Ellen Gethner, Stan Wagon, and Brian Wick: A StrollThough the Gaussian Primes, Mathematical SciencesResearch Institute, Berkeley, CA.

[4] Wayne L. McDaniel: Perfect Gaussian integers, Universityof Missouri - St. Louis, MO, 1972.

[5] Aakash Paul, Saransh Sharma, Somjit Datta and Subhashis Majumder: Primality Testing of Gaussian Integers,International Conference on Emerging Trends in Computationaland Applied Mathematics, pp. 333-337, June,2014.

[6] Aakash Paul, Somjit Datta, Saransh Sharma and Subhashis Majumder: On the Use of Gaussian Integers inPublic Key Cryptosystems, 3rd International Conferenceon Advanced Computing, Networking, and Informatics(ICACNI), June 2015.

[7] William H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery: Numerical Recipes in C -The Art of Scientific Computing, Cambridge UniversityPress, pp 948-950, 1988.

[8] Legh Wilber Reid : The elements of the Theory ofAlgebraic Numbers, pp 194-195, 1910.

[9] Joseph H. Silverman: A Friendly introduction to Number Theory, Third edition, pp 239-269, 2009.

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Acknowledgements

• We would like to thank our mentors, Prof. (Dr.) Subhashis Majumder and Prof. (Dr.) Somjit Datta, without whom our first research paper would have remained unfathomable.

• Also, a word of gratitude to our parents for their continual support.

• Finally, to IEEE ICCC 2015, for making our dream a reality.

Thank You!

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