Le equazioni diofanteeax 4-bx 2+1=□, 4x 4+x 2 y 2+4y 4=□.

Summary Si determinano tre classi di quartiche u²=ax⁴+1, a e b interi, b²−4a≠□ per le quali l'appartenenza di un punto razionale (x₀,y₀),x₀≠0 porta l'esistenza nella stessa quartica di infiniti punti razionali. Si dimostra pure che l'equazione 4x²+x²y²+4y⁴=□ non ammette soluzioni intere con |xy| > 1. A BeniaminoSegre nel suo 70^mo compleanno. Entrata in Redazione il 26 febbraio 1973.     

Über die Diophantische Gleichungax 4+bx 2 y 2+cy 4=ez 2

Ohne ZusammenfassungEingereicht zur Erlangung des Grades eines Dr. phil. habil. an der Philosophischen Fakultat der Universität Leipzig.     

The equation $$x^4+2^ny^4=z^4$$ x 4 + 2 n y 4 = z 4 in algebraic number fields

Acta Mathematica Hungarica - Let n be a positive integer. We show that if the equation $$(1) \qquad \qquad \qquad x^4+2^ny^4=z^4$$ has a solution (x,y,z) in a cubic number field K with $$xyz \neq...     

On the diophantine equation y2 = 4qn + 4q + 1

It is known that a certain class of [n, k] codes over GF(q) is related to the diophantine equation y² = 4qⁿ + 4q + 1 (1). In Parts I and II of this paper, two different, and in a certain sense complementary, methods of approach to (1) are discussed and some results concerning (1) are given as applications. A typical result is that the only solutions to (1) are (y, n) = (5, 1), (7, 2), (11, 3) when q = 3 and (y, n) = (2q + 1, 2) when q = 3^f, f >- 2.     

Die Diophantische Gleichungmx 2+ny2=z4

Ohne Zusammenfassung     

关于丢番图方程X~2+4Y~4=pZ~4

设p为素数,p=4A~2+1+2|A,A∈N~*.运用二次和四次丢番图方程的结果证明了方程G:X~2+4Y~4=pZ~4,gcd(X,Y,Z)=1,除开正整数解(X,Y,Z)=(1,A,1)外,当A≡1(mod4)时,至多还有正整数解(X,Y,Z)满足X=|p(a~2-b~2)~2-4(A(a~2-b~2)±ab)~2|,Y~2=A(a~2-b~2)~2±2ab(a~2-b~2)-4a~2b~2A,Z=a~2+b~2;当A≡3(mod4)时,至多还有正整数解(X,Y,Z)满足X=|4a~2b~2A-(4abA±(a~2-b~2))~2|,Y~2=4a~2b~2A±2ab(a~2-b~2)-A(a~2-b~2)~2,Z=a~2+b~2.这里a,b∈N~*并且ab,gcd(a,b)=1,2|(a+b).同时具体给出了p=5时方程G的全部正整数解.     

The diophantine equation x2 = 4qa2 + 4q + 1, with an application to coding theory

All solutions in positive integers of the equation of the title are found, under the restriction that q be a prime power. A method of F. Beukers is used to show that q is bounded, apart from trivial solutions. The remaining q′s are dealt with by congruence arguments and by a method using second-order recurrence sequences. An application is given toward the classification of certain [n,k] codes over GF(q).     

The al-husayn equation x 4 + y 2 = z 2

We study the set of all natural solutions of the equation x ⁴ + y ² = z ², obtain general formulas describing all such solutions, and prove their equivalence.     

A simple method for solving the diophantine equation Y2 = X4 + aX3 + bX2 + cX + d

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A simple method for solving the diophantine equation Y2 = X4 + aX3 + bX2 + cX + d

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The exponential Diophantine equation x2 + (3n 2 + 1) y = (4n 2 + 1) z

Let n be a positive integer. In this paper, using the results on the existence of primitive divisors of Lucas numbers and some properties of quadratic and exponential diophantine equations, we prove that if n ≡ 3 (mod 6), then the equation x ² + (3n ² + 1) ^y = (4n ² + 1) ^z has only the positive integer solutions (x, y, z) = (n, 1, 1) and (8n ³ + 3n, 1, 3).     

SBIBD(4k 2, 2k 2 +k,k 2 +k) and Hadamard matrices of order 4k 2 with maximal excess are equivalent

We show that anSBIBD(4k ², 2k ² +k,k ² +k) is equivalent to a regular Hadamard matrix of order 4k ² which is equivalent to an Hadamard matrix of order 4k ² with maximal excess.We find many newSBIBD(4k ², 2k ² +k,k ² +k) including those for evenk when there is an Hadamard matrix of order 2k (in particular all 2k 210) andk {1, 3, 5,..., 29, 33,..., 41, 45, 51, 53, 61,..., 69, 75, 81, 83, 89, 95, 99, 625, 3^2m , 253^2m ,m 0}.     

Electron paramagnetic resonance of Mn2+ in (NH4)2SO4 single crystal

Electron paramagnetic resonance investigation of Mn²⁺ in (NH₄)₂SO₄ single crystal is discussed both in paraelectric and ferroelectric phases of the crystal. Mn²⁺ is found to substitute one of the two possible types (α andβ) of NH ₄ ⁺ ions and get associated with the second type of NH ₄ ⁺ vacancy, the vacancy being the second distant neighbour in thebc-plane. As The line joining Mn²⁺ substituted NH ₄ ⁺ site and NH ₄ ⁺ vacancy lies at an angle of 18° from the crystallographicb-axis in thebc-plane. As the temperature is lowered to ? 56° C the crystal becomes ferroelectric and the spectrum in the paraelectric phase splits into two from which it appears that two sets of Mn²⁺ sites which are magnetically equivalent in the paraelectric phase become inequivalent in the ferroelectric phase. The spin Hamiltonian analysis is presented for the spectrum in the paraelectric phase.     

On the hypersurfacex+x 2y+z 2+t 3=0 in ℂ4 or a ℂ4-like threefold which is not C3

In this note it will be proved that the threefold in ?⁴ which is given byx+x ² y+z ²+t ³=0 is not isomorphic to ?³. Here ? is the field of complex numbers.     

Self-converse Mendelsohn designs with block size 4t + 2

A Mendelsohn design MD(v, k, λ) is a pair (X, B) where X is a v-set together with a collection B of cyclic k-tuples from X such that each ordered pair from X, as adjacent entries, is contained in exactly λk-tuples of B. An MD(v, k, λ) is said to be self-converse, denoted by SCMD(v, k, λ) = (X, B, f), if there is an isomorphic mapping from (X, B) to (X, B⁻¹), where B⁻¹ = {B⁻¹ = 〈x_k, x_k−1, … x₂, x₁〉; B = 〈x₁, … ,x_k〉 ∈ B.}. The existence of SCMD(v, 3, λ) and SCMD(v, 4, 1) has been settled by us. In this article, we will investigate the existence of SCMD(v, 4t + 2, 1). In particular, when 2t + 1 is a prime power, the existence of SCMD(v, 4t + 2, 1) has been completely solved, which extends the existence results for MD(v, k, 1) as well. © 1999 John Wiley & Sons, Inc. J. Combin Designs 7: 283–310, 1999     

Kadomtsev–Petviashvili Equation Revisited and Integrability in 4 + 2 and 3 + 1

The Cauchy problem of Kadomtsev–Petviashvili I (KPI) was reduced to a nonlocal Riemann–Hilbert (RH) problem by the author and Ablowitz in 1983. This formulation was based on the introduction of two spectral functions (nonlinear Fourier transforms, FTs). This formalism was improved by Boiti et al. [ 1 ], where it was shown that the earlier nonlocal RH problem can be formulated in terms of a single spectral function (nonlinear FT). A different formalism was presented by Zhou [ 2 ], where the Cauchy problem was rigorously solved in terms of a linear integral equation involving a nonanalytic eigenfunction. Here, we first revisit the above results and then review some recent results about the derivation of integrable generalizations of KP in 4 + 2 (i.e., in four spatial and two temporal dimensions), as well as in 3 + 1 (i.e., in three spatial and one temporal dimensions).     

丢番图方程X~2-(a~2+4p~(2n))Y~4=-4p~(2n)

令α,n≥1为整数,p为素数.本文证明了丢番图方程X~2-(a~2+4p~(2n))Y~4=-4p~(2n)以及X~2-(a~2+p~(2n))Y~4=-p~(2n)在一定条件下最多只有两组互素的正整数解(X,Y).