A. Frempong - Academia.edu (original) (raw)
Papers by A. Frempong
viXra, 2020
An ingenious proof of Fermat's Last theorem has been covered in this paper. Fermat's Last... more An ingenious proof of Fermat's Last theorem has been covered in this paper. Fermat's Last theorem states that if A, B, C, n are positive integers; A, B, and C are coprime, and n > 2, then the equation A^n + B^n = C^n, has no solutions. The principles applied in the proof are based on the properties of the factored Fermat's equation. The main principle for obtaining relationships between the prime factors on the left side of the equation and the prime factor on the right side of the equation is that the greatest common power of the prime factors on the left side of the equation is the same as a power of the prime factor on the right side of the equation. High school students can learn and prove this theorem as a bonus question on a final class exam.
viXra, 2015
This paper covers the solutions of the Euler equations in 3-D and 4-D for incompressible fluid fl... more This paper covers the solutions of the Euler equations in 3-D and 4-D for incompressible fluid flow. The solutions are the spin-offs of the author's previous analytic solutions of the Navier-Stokes equations (vixra:1405.0251 of 2014). However, some of the solutions contained implicit terms. In this paper, the implicit terms have been expressed explicitly in terms of x, y, z and t. The author applied a new law, the law of definite ratio for fluid flow. This law states that in incompressible fluid flow, the other terms of the fluid flow equation divide the gravity term in a definite ratio, and each term utilizes gravity to function. The sum of the terms of the ratio is always unity. This law evolved from the author's earlier solutions of the Navier-Stokes equations. In addition to the usual approach of solving these equations, the Euler equations have also been solved by a second method in which the three equations in the system are added to produce a single equation which is ...
This study examines the extent to which farmers trained by the Food Research Institute have taken... more This study examines the extent to which farmers trained by the Food Research Institute have taken up mushroom farming and also determines the profitability of mushroom production. Specifically, the study describes trends in the levels of mushroom production since 1995, the rate of adoption and dis-adoption of mushroom farming by trainees, the profitability of mushroom cultivation and identifies constraints to mushroom farming from the perspective of farmers. Trend description involves annual output of respondents in the Greater Accra region from 1995 to 1999. The profitability of mushroom cultivation was determined by the use of the Benefit-Cost Ratio ,Net Present Value and the Internal Rate of Return criteria. The constraints to mushroom cultivation from the perspective of the farmers were listed and ranked in decreasing order of importance. Results from the study reveal that there is an increasing trend in production by the individual farmers in the Greater Accra region, and that ...
viXra, 2017
Coincidences. The US Supreme Court consists of nine members, one of whom is the Chief Justice of ... more Coincidences. The US Supreme Court consists of nine members, one of whom is the Chief Justice of the Court. So also, a one-direction Navier-Stokes equation consists of nine members, one of which is the indispensable gravity term, without which there would be no incompressible fluid flow as shown by the solutions of the N-S equations (viXra:1512.0334). Another coincidence is that numerologically, the number, 9, is equivalent to the 1800's (1 + 8 + 0 + 0 = 9) time period during which the number of the members of the Supreme Court became fixed at 9, while the formulation of the nine-term N-S equations was completed. Also, another coincidence is that the solutions of the N-S equations were completed (viXra:1512.0334) by the author in the year, 2016 (2 + 0 +1+ 6 = 9). Using a new introductory approach, this paper covers the author's previous solutions of the N-S equations (viXra:1512.0334). In particular, the N-S solutions have been compared to the equations of motion and liquid ...
viXra, 2014
Assuming the sum of the original Riemann series is L, a ratio method was used to split-up the ser... more Assuming the sum of the original Riemann series is L, a ratio method was used to split-up the series equation into sub-equations and each sub-equation was solved in terms of L, and ratio terms. It is to be noted that unquestionably, each term of the series equation contributes to the sum, L, of the series. There are infinitely many sub-equations and solutions corresponding to the infinitely many terms of the series equation. After the sum, L, and the ratio terms have been determined and substituted in the corresponding equations, the Riemann hypothesis would surely be either proved or disproved, since the original equation is being solved. Solving the original series equation eliminates possible hidden flaws in derived equations and consequent solutions.
viXra, 2020
In a page margin, the author proves directly the original Beal conjecture that if A^x + B^y = C^z... more In a page margin, the author proves directly the original Beal conjecture that if A^x + B^y = C^z where A, B, C, x. y, z are positive integers and x, y, z > 2, then A, B, and C have a common prime factor. The principles applied in the proof are based on the properties of the factored Beal equation. The main principle for obtaining relationships between the prime factors on the left side of the equation and the prime factor on the right side of the equation is that the greatest common power of the prime factors on the left side of the equation is the same as a power of the prime factor on the right side of the equation.
viXra, 2020
The author proves both the original Beal conjecture and the equivalent Beal conjecture. The origi... more The author proves both the original Beal conjecture and the equivalent Beal conjecture. The original Beal conjecture states that if A^x + B^y = C^z, where A, B, C, x, y, z are positive integers and x, y, x >2 then A, B, and C have a common prime factor. The equivalent Beal conjecture states that if A, B, C, x, y, z are positive integers and A, B, and C are coprime , and x, y, z >2, then the equation if A^x + B^y = C^z, has no solutions. The principles applied in both proofs are based on the same properties of the factored Beal equation. However the proof of the equivalent conjecture is by contradiction. The main principle for obtaining relationships between the prime factors on the left side of the equation and the prime factor on the right side of the equation is that the greatest common power of the prime factors on the left side of the equation is the same as a power of the prime factor on the right side of the equation. High school students can learn and prove this conject...
Pediatric Neurosurgery, 1997
Six patients presented with either entrapped fourth ventricles or noncommunicating cerebrospinal ... more Six patients presented with either entrapped fourth ventricles or noncommunicating cerebrospinal fluid collections of the posterior fossa requiring drainage. These collections were treated with shunt systems whose proximal catheter was placed into the fourth ventricle via a coronal burr hole using an endoscope guided by Eleckta's ISG Viewing Wand. The technique and its advantages are described as are the complications and early outcomes.
viXra, 2016
Beal conjecture has been proved on a single page; and the proof has been specialized to prove Fer... more Beal conjecture has been proved on a single page; and the proof has been specialized to prove Fermat's last theorem, on half of a page. The approach used in the proof is exemplified by the following system. If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one would first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solutions for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2, will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 ...
viXra, 2020
An ingenious proof of Fermat's Last theorem has been covered in this paper. Fermat's Last... more An ingenious proof of Fermat's Last theorem has been covered in this paper. Fermat's Last theorem states that if A, B, C, n are positive integers; A, B, and C are coprime, and n > 2, then the equation A^n + B^n = C^n, has no solutions. The principles applied in the proof are based on the properties of the factored Fermat's equation. The main principle for obtaining relationships between the prime factors on the left side of the equation and the prime factor on the right side of the equation is that the greatest common power of the prime factors on the left side of the equation is the same as a power of the prime factor on the right side of the equation. High school students can learn and prove this theorem as a bonus question on a final class exam.
viXra, 2015
This paper covers the solutions of the Euler equations in 3-D and 4-D for incompressible fluid fl... more This paper covers the solutions of the Euler equations in 3-D and 4-D for incompressible fluid flow. The solutions are the spin-offs of the author's previous analytic solutions of the Navier-Stokes equations (vixra:1405.0251 of 2014). However, some of the solutions contained implicit terms. In this paper, the implicit terms have been expressed explicitly in terms of x, y, z and t. The author applied a new law, the law of definite ratio for fluid flow. This law states that in incompressible fluid flow, the other terms of the fluid flow equation divide the gravity term in a definite ratio, and each term utilizes gravity to function. The sum of the terms of the ratio is always unity. This law evolved from the author's earlier solutions of the Navier-Stokes equations. In addition to the usual approach of solving these equations, the Euler equations have also been solved by a second method in which the three equations in the system are added to produce a single equation which is ...
This study examines the extent to which farmers trained by the Food Research Institute have taken... more This study examines the extent to which farmers trained by the Food Research Institute have taken up mushroom farming and also determines the profitability of mushroom production. Specifically, the study describes trends in the levels of mushroom production since 1995, the rate of adoption and dis-adoption of mushroom farming by trainees, the profitability of mushroom cultivation and identifies constraints to mushroom farming from the perspective of farmers. Trend description involves annual output of respondents in the Greater Accra region from 1995 to 1999. The profitability of mushroom cultivation was determined by the use of the Benefit-Cost Ratio ,Net Present Value and the Internal Rate of Return criteria. The constraints to mushroom cultivation from the perspective of the farmers were listed and ranked in decreasing order of importance. Results from the study reveal that there is an increasing trend in production by the individual farmers in the Greater Accra region, and that ...
viXra, 2017
Coincidences. The US Supreme Court consists of nine members, one of whom is the Chief Justice of ... more Coincidences. The US Supreme Court consists of nine members, one of whom is the Chief Justice of the Court. So also, a one-direction Navier-Stokes equation consists of nine members, one of which is the indispensable gravity term, without which there would be no incompressible fluid flow as shown by the solutions of the N-S equations (viXra:1512.0334). Another coincidence is that numerologically, the number, 9, is equivalent to the 1800's (1 + 8 + 0 + 0 = 9) time period during which the number of the members of the Supreme Court became fixed at 9, while the formulation of the nine-term N-S equations was completed. Also, another coincidence is that the solutions of the N-S equations were completed (viXra:1512.0334) by the author in the year, 2016 (2 + 0 +1+ 6 = 9). Using a new introductory approach, this paper covers the author's previous solutions of the N-S equations (viXra:1512.0334). In particular, the N-S solutions have been compared to the equations of motion and liquid ...
viXra, 2014
Assuming the sum of the original Riemann series is L, a ratio method was used to split-up the ser... more Assuming the sum of the original Riemann series is L, a ratio method was used to split-up the series equation into sub-equations and each sub-equation was solved in terms of L, and ratio terms. It is to be noted that unquestionably, each term of the series equation contributes to the sum, L, of the series. There are infinitely many sub-equations and solutions corresponding to the infinitely many terms of the series equation. After the sum, L, and the ratio terms have been determined and substituted in the corresponding equations, the Riemann hypothesis would surely be either proved or disproved, since the original equation is being solved. Solving the original series equation eliminates possible hidden flaws in derived equations and consequent solutions.
viXra, 2020
In a page margin, the author proves directly the original Beal conjecture that if A^x + B^y = C^z... more In a page margin, the author proves directly the original Beal conjecture that if A^x + B^y = C^z where A, B, C, x. y, z are positive integers and x, y, z > 2, then A, B, and C have a common prime factor. The principles applied in the proof are based on the properties of the factored Beal equation. The main principle for obtaining relationships between the prime factors on the left side of the equation and the prime factor on the right side of the equation is that the greatest common power of the prime factors on the left side of the equation is the same as a power of the prime factor on the right side of the equation.
viXra, 2020
The author proves both the original Beal conjecture and the equivalent Beal conjecture. The origi... more The author proves both the original Beal conjecture and the equivalent Beal conjecture. The original Beal conjecture states that if A^x + B^y = C^z, where A, B, C, x, y, z are positive integers and x, y, x >2 then A, B, and C have a common prime factor. The equivalent Beal conjecture states that if A, B, C, x, y, z are positive integers and A, B, and C are coprime , and x, y, z >2, then the equation if A^x + B^y = C^z, has no solutions. The principles applied in both proofs are based on the same properties of the factored Beal equation. However the proof of the equivalent conjecture is by contradiction. The main principle for obtaining relationships between the prime factors on the left side of the equation and the prime factor on the right side of the equation is that the greatest common power of the prime factors on the left side of the equation is the same as a power of the prime factor on the right side of the equation. High school students can learn and prove this conject...
Pediatric Neurosurgery, 1997
Six patients presented with either entrapped fourth ventricles or noncommunicating cerebrospinal ... more Six patients presented with either entrapped fourth ventricles or noncommunicating cerebrospinal fluid collections of the posterior fossa requiring drainage. These collections were treated with shunt systems whose proximal catheter was placed into the fourth ventricle via a coronal burr hole using an endoscope guided by Eleckta's ISG Viewing Wand. The technique and its advantages are described as are the complications and early outcomes.
viXra, 2016
Beal conjecture has been proved on a single page; and the proof has been specialized to prove Fer... more Beal conjecture has been proved on a single page; and the proof has been specialized to prove Fermat's last theorem, on half of a page. The approach used in the proof is exemplified by the following system. If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one would first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solutions for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2, will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 ...