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गणिनी आर्यिकारत्न श्री ज्ञानमती अभिवन्दन ग्रंथ
.
Uz
A positive integer N is called a perfect number if it is the sum of its proper divisors (including 1). Thus 6 is a perfect number because it is the sum of its proper divisors 1, 2 and 3.
The next perfect number is 28 because 1 + 2 + 4 + 7 + 14 = 28. In fact it is known that the numbers P, given by (1) will be perfect whenever the factor
fn = (21 - 1) is a prime number (that is, a number which has no proper divisor except 1). Table II depicts values of f, for some n.
TABLE III n = 1 2 3 4 5 6 7 8 fo = 1 3 7 15 31 63 127 255.
it can be checked that the numbers 3, 7, 31 and 127 are primes. Hence P, will be perfect for the corresponding values of n as mentioned in Table I. Also it should be noticed that, starting witht the Jambudvipa itself, the numbers f, are nothing but the successive sums of the widths (in lakhs yojanas) of the various islands and seas as one crosses them in one direction.
Thus the discussion of ancient Jaina cosmography and the related calculations involve not only sums of geometrical progression but also examples of perfect numbers.
It is said that the first four perfect numbers were also known to the ancient Greeks. The fifth perfect number corresponds to n = 13 for which f13 = 213 - 1
= 8191 which is prime,
and
P13 = 213 – 1). 212
= 3355, 0336 which is perfect. The next three perfect numbers will come by taking n = 17, 19, and 31, (the last of which will give a perfect number consisting of 19 digits).
In 1757 the great Swiss Mathematician Leonhard Euler proved mathematically that every even perfect number must be in the form P. No other type of perfect number has ever been found. We can say almost safely that all perfect numbers represent the number of Khandas (in terms of those of Lavana Sea) of some Island or Sea in Jaina cosmography in which the number of rings of such Islands and Seas are stated to be asamkhyåta ('unenumerable').
REFERENCES AND NOTES: 1. Tilovapannatti Chapter V. gåthá 32, states that the diameter of the Jambů Island is one lakh yojanas and the viskambha (vistara or width) of rings from Lavana Sea to Svayambhuramana Sea are successively double (each time), that is,
W. = 2W-1, n = 1, 2, 3, .... We are using the new edition by C.P. Patni of the text which is published along with the Hindi translation of Aryikā Visud
dhamatí, Kota, 1984 (vol. I), 1986 (vol. II) and 1988 (vol. III). For V, 32, see vol. II, p.7. 2. L.C. Jain, 'Mathematics of Tiloyapannatti in Hindi), essay attached to Jambudiva Pannatti Sangaho (Sholapur, 1958), intro
ductory pp. 69-70. It should be noted that the factor 9 in formula (11) is the value of 37, and not of ? 3. Şarkhandägama (with Dhavala) edited with Hindi translation by Hiralal Jain, Vol. IV, Amraoti, 1942, pp. 195-196. 4. M.L. Nankar, "History of Perfect Numbers", Ganita Bharati, Vol. I (1979), pp. 7-8. It may be pointed out that (see Table
1) although 120 is not perfect, it is a multiply-perfect number because the sum of its proper divisors. 1+2+3+4+5+6+8+10+12+15+20+24+30+40+60 = 2 x 120 Also, 672 is another such number
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