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Section 4. Mathematics
Section 4. Mathematics
Drushinin Victor Vladimirinich, National research nuclear University “MEPHI”, Sarov physical-technical Institute E-mail: Sarov, vvdr@newmail.ru
Nonlinear diophantine equation to generate prime numbers
Abstract: We have created a new non-linear Diophantine equation with two unknowns for the generation of large prime numbers. We gave examples of receipt of such numbers.
Keywords: primes, the Diophantine equation.
Consider a primes n-th floor of the form: p(n, m) = (2m)2 +1, where m and n are natural numbers. Divisors for composite number of this floor s (n, m) of the same kind are primes p (n, p) = 2n+1 p +1 of auxiliary floor. Indeed, the equality s (n, m) = p (n, p1 )p (n +1, p2 ) isnot possible under the law of conservation of parity. Therefore, the primes p(n, p) are the best result of the decomposition of s (n, m) at multipliers. For example:
p(2,4) = 4097 = 17 • 241 = p (2,2) • p (2,30). The presence of two classes of primes makes it easier factorization problem and the generation of large prime numbers, i. e. in the canonical decomposition of s (n, m) includes a limited
number of primes of the form p (n, p). It is mathematically written as:
(2
m )2 + 1 =
N ,
n(+i +1
(1)
If the product in (1) degenerates into one multiplier, then the left is prime with a good index m and where in
p = 2
2" -n-1 2"
. Equation (1) is the first simplification for the
task of generating large primes.
The second simplification is to use the determinant criteria for divisibility [1], which allows to quickly check the divisibility of any two numbers on each other. This feature divisibility was proposed by the author of this article and is as follows. There are dividend A = ax ■ ßk + a0, and the divisor B = b ■ ßL + b0, i. e. multiplier ß is present in both composition of numbers in different degrees. This allows to create function of Druzhinin’s divisibility:
D (A; B ) =
(-1))+1 b b
= ab + (-1) b[a0
(2)
If D (A; B) has a multiple of B, it is necessary and sufficient that A is a multiple of B. Since the factor ß is removed from D(A;B), then in many cases D(A;B) A, that mak-
ing it easier to check divisibility. Here is an example on using this sign of divisibility. There are :
A = 50011 = 5-104 +11,B = 13 = 1-10 + 3. According to (2)
D (0011;13) = 5 • 34 +11 = 416 = 13 • 32. Hence it follows that A = 50011 is multiple of B = 13. Indeed, 50011 = 13 • 3847.
We apply determinate sign of divisibility for two classes of numbers and denote:
2n = a(n + l) + Y, a1 = 2Y mn, ß = 2, a0 = 1,
b1 = p, L = (n +1), k = a, b0 = 1.
Substituting these values in (2), we obtain the sign of divisibility:
2Y m2n 1
D
s (n, m);p (n, p)
i-\y+lp 1
= 2ymr +(-l)“p (3)
Since we are interested in the expansion of the composite number of n -th floor s(n, m)2 = (2m)2n +1 on primes for
of the auxiliary floor p (n, p) ,then we investigate divisibility D(n,m) for p (n, p),i. e. we compile Diophantine equation:
2y mr +(-l )apa= x (n+1 p +1), (4)
where x is an integer. This is the required non-linear Diophantine equation to find a primes of the n -th floor.
To solve this problem for a fixed n, we need first create a set of primes for auxiliary floor, i. e. we need define a set of a god indices p giving primes p (n, p). After that, we set a specific number m and solve the equation (4) for the unknown x and p. Thus it is enough verify that x is an integer, while it’s not specifying. The resulting set of values {p,} gives primes for decomposition s (n, m). If there are the only p and 2n+1 p +1 = s (n, m) in the solution (4), the number s (n, m) is a primes of n -th floor.
Consider the work of the scheme by specific examples.
1. Let n = 1. There is s (n, m) = 4m2 +1,p(n, p) = 4p +1. Equation (4) has the form m2 — p = x(p +l) If m = 1, we have one solution p = 1, x = 0. This gives the first prime of the first floor p(1,1) = 5. When m = 2 we have also one decision p = 4, x = 0 and obtain the second floor of the first prime p (l,2) = 17. m = 3 gives the third prime p (1,3) = 37. When m = 4 there are two solutions of the equation (4): p1 = 1 and
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Nonlinear diophantine equation to generate prime numbers
p 2 = 3. We got the first composite number of the first floor 5(1,4) = 65 = 5-13 . When m = 33 we have one solution p = 1089, which gives another prime of the first floor p (1,33) = 4357.
2. Let n = 2. There is s (2, m) = 16m4 +1,p (2, p) = 8p +1. Equation (4) has the form 2m4 - p = x (p +1) If m = 1, we have one solution p = 2 and we get the first prime of the second floor p (2,1) = 17 . When m = 4 we have two solution p1 = 2 and p2 = 30. We got the first composite number of the second floor s (2,4) = 4097 = 17-241. The index m = 28
gives one solution p = 1229312, i. e. we get another prime of the second floor p (2,28) = 9834497.
3. Let us find the decomposition of Fermaof fifth floor s (5,l) = 4294967297 . Our scheme leads to the equation 4 -p = x(б4p +1). When p = 10 we have the equality (4 -100000) = -156 • 641, i. e., the fifth number of Farm is composite. It was first discovered by Euler.
The solution of equation (3) greatly reduces operation of factorization and search of a primes, and apparently could provide new record primes.
The author thanks A. Lazarev for assistance in calculating large prime numbers under this scheme, as well as Professor, doctor physical-mathematical sciences Shevyahov N. S. for interest in the work and valuable comments.
References:
1. Druzhinin V V The Determinant Criteria for Divisibility, - 2012, - Sarov, Russia, - P. 24.
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