Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ...Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ ∗ , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: ( A α ) 2 + ( B β ) 2 = ( C γ ) 2 , such that its area Δ= 1 2 A α B β =n;or in an equivalent way, to that of the existence of numbers U 2 , V 2 , W 2 ∈ ℚ 2∗ that are in an arithmetic progression of reason n;Problem equivalent to the existence of: ( a,b,c )∈ ℕ 3∗ prime in pairs, and f∈ ℕ ∗ , such that: ( a−b 2f ) 2 , ( c 2f ) 2 , ( a+b 2f ) 2 are in an arithmetic progression of reason n;And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a 2 + b 2 = c 2 , such that its area Δ= 1 2 ab=n f 2 , where f∈ ℕ ∗ , and this last equation can be written as follows, when using Pythagorician divisors: (1) Δ= 1 2 ab= 2 S−1 d e ¯ ( d+ 2 S−1 e ¯ )( d+ 2 S e ¯ )=n f 2;Where ( d, e ¯ )∈ ( 2ℕ+1 ) 2 such that gcd( d, e ¯ )=1 and S∈ ℕ ∗ , where 2 S−1 , d, e ¯ , d+ 2 S−1 e ¯ , d+ 2 S e ¯ , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When n=1 , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers n=1 (resp. n=2 ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers n=p≡a ( ( mod8 ) , gcd( a,8 )=1 ). To prove it, we use a classical proof by induction on k , that shows the non-solvability property of any of the following systems ( t=0 , corresponding to case n=1 (resp. t=1 , corresponding to case n=2 )): ( Ξ t,k ){ X 2 + 2 t ( 2 k Y ) 2 = Z 2 X 2 + 2 t+1 ( 2 k Y ) 2 = T 2 , where k∈ℕ;and solutions ( X,Y,Z,T )=( D k , E k , f k , f ′ k )∈ ( 2ℕ+1 ) 4 , are given in pairwise prime numbers.2020-Mathematics Subject Classification 11A05-11A07-11A41-11A51-11D09-11D25-11D41-11D72-11D79-11E25 .展开更多
Attiéké is fermented cassava semolina steamed and consumed with proteins and vegetables. From Ivorian traditional origin, its popularity is increasing in Africa and production networks are developing. Howeve...Attiéké is fermented cassava semolina steamed and consumed with proteins and vegetables. From Ivorian traditional origin, its popularity is increasing in Africa and production networks are developing. However, the growth of attiéké industry is still curbed by lack of technical skills of producers and irregular quality, depending on cassava variety, processing and inoculum. In the present study, a traditional inoculum and four cassava varieties (two improved local (IAC and Bonoua) and two improved sweet ones from Nigeria (Olekanga and TMS 4 (2) 1425)) were used for producing attiéké Ebrié. For the sensorial evaluation of prepared attiéké, qualitative sensory profiling was used. The sensory quality descriptors where quantitatively measured by a trained sensory panel. With this method the most important quality criteria of attiéké were judged. These quality criteria were selected with the aid of surveys on its production and consumption. The relationships between sensory attributes of the four attiéké prepared of the different cassava varieties and biochemical characteristics were studied. Sensory analysis revealed that the four attiéké were cream-colored products with sweet and/or sour tastes, made of cohesive and well-formed grains of different sizes, with an odor of fermentation specific to attiéké and showing a firm texture. The differences between sensory profiles of the four attiéké produced from four cassava varieties (12 months growth) concern only some descriptors of quality. Pearson correlation coefficients between physicochemical descriptors of quality showed that biochemical parameters may help predict organoleptic characteristics of attiéké Ebrié (pH-size of grains: r = 0.99;pH-acidity: r = ?0.92;acidity-rounded grains: r = ?0.98: starch-cohesion between grains: r = ?0.96;starch-sweet: r = ?0.95;starch-granulous: r = 0.97, total sugar-yellow color: r = 0.96;total sugar-aroma of attiéké: r = 0.96;reducing sugar-odor of attiéké: r = 0.95;cyanide-fibrous: r = 0.95).展开更多
The goal of this paper is to show that there are infinitely many number fields K/Q, for which there is no inert prime p ∈ N*, i.e. ∀p ∈ N* a prime number, prime ideal of K such that where: Zk</sub> i...The goal of this paper is to show that there are infinitely many number fields K/Q, for which there is no inert prime p ∈ N*, i.e. ∀p ∈ N* a prime number, prime ideal of K such that where: Zk</sub> is the Dedekind domain of the integer elements of K. To prove such a result, consider for any prime p, the decomposition into a product of prime ideals of Zk</sub>, of the ideal . From this point, we use on the one hand: 1) The well- known property that says: If , then the ideal pZ<sub>k</sub> decomposes into a product of prime ideals of Zk</sub> as following: . (where:;is the irreducible polynomial of θ, and, is its reduction modulo p, which leads to a product of irreducible polynomials in Fp[X]). It is clear that because if is reducible in Fp[X], then consequently p is not inert. Now, we prove the existence of such p, by proving explicit such p as follows. So we use on the other hand: 2) this property that we prove, and which is: If , is an irreducible normalized integer polynomial, whose splitting field is , then for any prime number p ∈ N: is always a reducible polynomial. 3) Consequently, and this closes our proof: let’s consider the set (whose cardinality is infinite) of monogenic biquadratic number fields: . Then each f<sub>θ</sub>(X) checks the above properties, this means that for family M, all its fields, do not admit any inert prime numbers p ∈ N. 2020-Mathematics Subject Classification (MSC2020) 11A41 - 11A51 - 11D25 - 11R04 - 11R09 - 11R11 - 11R16 - 11R32 - 11T06 - 12E05 - 12F05 -12F10 -13A05-13A15 - 13B02 - 13B05 - 13B10 - 13B25 -13F05展开更多
文摘Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ ∗ , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: ( A α ) 2 + ( B β ) 2 = ( C γ ) 2 , such that its area Δ= 1 2 A α B β =n;or in an equivalent way, to that of the existence of numbers U 2 , V 2 , W 2 ∈ ℚ 2∗ that are in an arithmetic progression of reason n;Problem equivalent to the existence of: ( a,b,c )∈ ℕ 3∗ prime in pairs, and f∈ ℕ ∗ , such that: ( a−b 2f ) 2 , ( c 2f ) 2 , ( a+b 2f ) 2 are in an arithmetic progression of reason n;And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a 2 + b 2 = c 2 , such that its area Δ= 1 2 ab=n f 2 , where f∈ ℕ ∗ , and this last equation can be written as follows, when using Pythagorician divisors: (1) Δ= 1 2 ab= 2 S−1 d e ¯ ( d+ 2 S−1 e ¯ )( d+ 2 S e ¯ )=n f 2;Where ( d, e ¯ )∈ ( 2ℕ+1 ) 2 such that gcd( d, e ¯ )=1 and S∈ ℕ ∗ , where 2 S−1 , d, e ¯ , d+ 2 S−1 e ¯ , d+ 2 S e ¯ , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When n=1 , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers n=1 (resp. n=2 ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers n=p≡a ( ( mod8 ) , gcd( a,8 )=1 ). To prove it, we use a classical proof by induction on k , that shows the non-solvability property of any of the following systems ( t=0 , corresponding to case n=1 (resp. t=1 , corresponding to case n=2 )): ( Ξ t,k ){ X 2 + 2 t ( 2 k Y ) 2 = Z 2 X 2 + 2 t+1 ( 2 k Y ) 2 = T 2 , where k∈ℕ;and solutions ( X,Y,Z,T )=( D k , E k , f k , f ′ k )∈ ( 2ℕ+1 ) 4 , are given in pairwise prime numbers.2020-Mathematics Subject Classification 11A05-11A07-11A41-11A51-11D09-11D25-11D41-11D72-11D79-11E25 .
文摘Attiéké is fermented cassava semolina steamed and consumed with proteins and vegetables. From Ivorian traditional origin, its popularity is increasing in Africa and production networks are developing. However, the growth of attiéké industry is still curbed by lack of technical skills of producers and irregular quality, depending on cassava variety, processing and inoculum. In the present study, a traditional inoculum and four cassava varieties (two improved local (IAC and Bonoua) and two improved sweet ones from Nigeria (Olekanga and TMS 4 (2) 1425)) were used for producing attiéké Ebrié. For the sensorial evaluation of prepared attiéké, qualitative sensory profiling was used. The sensory quality descriptors where quantitatively measured by a trained sensory panel. With this method the most important quality criteria of attiéké were judged. These quality criteria were selected with the aid of surveys on its production and consumption. The relationships between sensory attributes of the four attiéké prepared of the different cassava varieties and biochemical characteristics were studied. Sensory analysis revealed that the four attiéké were cream-colored products with sweet and/or sour tastes, made of cohesive and well-formed grains of different sizes, with an odor of fermentation specific to attiéké and showing a firm texture. The differences between sensory profiles of the four attiéké produced from four cassava varieties (12 months growth) concern only some descriptors of quality. Pearson correlation coefficients between physicochemical descriptors of quality showed that biochemical parameters may help predict organoleptic characteristics of attiéké Ebrié (pH-size of grains: r = 0.99;pH-acidity: r = ?0.92;acidity-rounded grains: r = ?0.98: starch-cohesion between grains: r = ?0.96;starch-sweet: r = ?0.95;starch-granulous: r = 0.97, total sugar-yellow color: r = 0.96;total sugar-aroma of attiéké: r = 0.96;reducing sugar-odor of attiéké: r = 0.95;cyanide-fibrous: r = 0.95).
文摘The goal of this paper is to show that there are infinitely many number fields K/Q, for which there is no inert prime p ∈ N*, i.e. ∀p ∈ N* a prime number, prime ideal of K such that where: Zk</sub> is the Dedekind domain of the integer elements of K. To prove such a result, consider for any prime p, the decomposition into a product of prime ideals of Zk</sub>, of the ideal . From this point, we use on the one hand: 1) The well- known property that says: If , then the ideal pZ<sub>k</sub> decomposes into a product of prime ideals of Zk</sub> as following: . (where:;is the irreducible polynomial of θ, and, is its reduction modulo p, which leads to a product of irreducible polynomials in Fp[X]). It is clear that because if is reducible in Fp[X], then consequently p is not inert. Now, we prove the existence of such p, by proving explicit such p as follows. So we use on the other hand: 2) this property that we prove, and which is: If , is an irreducible normalized integer polynomial, whose splitting field is , then for any prime number p ∈ N: is always a reducible polynomial. 3) Consequently, and this closes our proof: let’s consider the set (whose cardinality is infinite) of monogenic biquadratic number fields: . Then each f<sub>θ</sub>(X) checks the above properties, this means that for family M, all its fields, do not admit any inert prime numbers p ∈ N. 2020-Mathematics Subject Classification (MSC2020) 11A41 - 11A51 - 11D25 - 11R04 - 11R09 - 11R11 - 11R16 - 11R32 - 11T06 - 12E05 - 12F05 -12F10 -13A05-13A15 - 13B02 - 13B05 - 13B10 - 13B25 -13F05
