We prove that there are at least (1) 4 primes between the squares of 2 consecutive odd primes and... more We prove that there are at least (1) 4 primes between the squares of 2 consecutive odd primes and (2) 1 prime between any two consecutive squares.
4. Find the values of sin 0 • , sin 90 • , sin 45 • . Bonus: find the values of sin 30 • and sin ... more 4. Find the values of sin 0 • , sin 90 • , sin 45 • . Bonus: find the values of sin 30 • and sin 60 • .
The Collatz function of a positive integer n is given by n/2 if n is even and 3n + 1 if n is odd.... more The Collatz function of a positive integer n is given by n/2 if n is even and 3n + 1 if n is odd. Lothar Collatz proposed in 1937 that repetitive iterations of this function, for any initial number n, will end as 1. For example, starting with n = 6, the iterations of the function are 6−3−10− 5−16−8−4−2−1. To prove the Collatz Conjecture, the Collatz function is examined for the binary representations of n. As opposed to popular trains of thought, a new perspective is introduced viewing the Collatz function as a series of string transformations as opposed to integer transformations. Algorithms are then created to iterate the Collatz function in binary as a string operations, which yield the generalisation that every starting number n can be reduced to 1 through repeated iteration of the Collatz function.
Albert Michelson's interferometer was an innovation into the physics of waves -finally an accurat... more Albert Michelson's interferometer was an innovation into the physics of waves -finally an accurate way was established to measure the wavelength of light. Michelson's apparatus was used in this experiment to measure the wavelength of light in an IF-HN05 laser. A laser was shone through a beam splitter, bouncing off onto two mirrors (one fixed, one movable) different distances from the beam splitter. When the beams of light reached back to the beam splitter, they interfered constructively. Because of this interference, an interference pattern of fringes occured on the screen and data could be collected. As the knob was turned causing the movable mirror to change distances from the beam splitter, new fringes appeared and were counted. Using the equation λ = 2d/n, where λ is wavelength of light in nm, d is distance the knob has turned in nm, and n is the number of fringes counted, data were collected for d and n. The data had a very high linear correlation (r = 99.99%, r 2 = 99.98%), and the regression line wasd = 167.9 + 313.6n (whered is predicted distance in nm). Since the slope of the regression line represents d/n, this regression line calculates that the wavelength of this light is 2 × 313.6nm = 627.2nm. Comparing the calculated value, 627.2nm, with the actual value, 632.8nm, yields a very small 0.88% percent error, due mainly to human handling of the precise interferometer. The high correlation between d and n deduces that d and n are in constant proportion, described by the equation λ = 2d/n, and that the wavelength of light produced by this laser is 632.8nm.
In 1897, J.J. Thomson became the first man to observe the charge to mass ratio using the apparatu... more In 1897, J.J. Thomson became the first man to observe the charge to mass ratio using the apparatus he designed, which is the same apparatus used in this experiment. Using a large magnetic field (created from the Helmholtz Coils), a current can be created from cathode rays (beams of electrons), which can be measured to find out the charge to mass ratio of the electron itself. Using classical variables, since mv 2 2
In 1644, Pietro Mengoli questioned the mathematical society about the sum of the reciprocals of t... more In 1644, Pietro Mengoli questioned the mathematical society about the sum of the reciprocals of the perfect square numbers. Thanks to Leonhard Euler in 1735, we now know that ∞ n=1
The relationship between math's two biggest constants, e and π, has been investigated by all of t... more The relationship between math's two biggest constants, e and π, has been investigated by all of the greats of mathematics -Euler, Bernoulli, et al. -but remains a haze today. 15-year-old Cody Dianopoulos speculates that the relationship between e and π is more than just that of Euler's great equation e iπ + 1 = 0, but that e and π have, in fact, a direct relationship. Dianopoulos goes into the approximations of their relationship and how computers can be used to confirm the values.
Why spend all of that time converting to component form and adding vectors when you can find the ... more Why spend all of that time converting to component form and adding vectors when you can find the magnitude of the resultant of two vectors with this new formula?
The Collatz Conjecture, first posed in 1937 by Lothar Collatz, has finally been confirmed through... more The Collatz Conjecture, first posed in 1937 by Lothar Collatz, has finally been confirmed through a series of nested proofs by fifteen-yearold Cody T. Dianopoulos.
We prove that there are at least (1) 4 primes between the squares of 2 consecutive odd primes and... more We prove that there are at least (1) 4 primes between the squares of 2 consecutive odd primes and (2) 1 prime between any two consecutive squares.
4. Find the values of sin 0 • , sin 90 • , sin 45 • . Bonus: find the values of sin 30 • and sin ... more 4. Find the values of sin 0 • , sin 90 • , sin 45 • . Bonus: find the values of sin 30 • and sin 60 • .
The Collatz function of a positive integer n is given by n/2 if n is even and 3n + 1 if n is odd.... more The Collatz function of a positive integer n is given by n/2 if n is even and 3n + 1 if n is odd. Lothar Collatz proposed in 1937 that repetitive iterations of this function, for any initial number n, will end as 1. For example, starting with n = 6, the iterations of the function are 6−3−10− 5−16−8−4−2−1. To prove the Collatz Conjecture, the Collatz function is examined for the binary representations of n. As opposed to popular trains of thought, a new perspective is introduced viewing the Collatz function as a series of string transformations as opposed to integer transformations. Algorithms are then created to iterate the Collatz function in binary as a string operations, which yield the generalisation that every starting number n can be reduced to 1 through repeated iteration of the Collatz function.
Albert Michelson's interferometer was an innovation into the physics of waves -finally an accurat... more Albert Michelson's interferometer was an innovation into the physics of waves -finally an accurate way was established to measure the wavelength of light. Michelson's apparatus was used in this experiment to measure the wavelength of light in an IF-HN05 laser. A laser was shone through a beam splitter, bouncing off onto two mirrors (one fixed, one movable) different distances from the beam splitter. When the beams of light reached back to the beam splitter, they interfered constructively. Because of this interference, an interference pattern of fringes occured on the screen and data could be collected. As the knob was turned causing the movable mirror to change distances from the beam splitter, new fringes appeared and were counted. Using the equation λ = 2d/n, where λ is wavelength of light in nm, d is distance the knob has turned in nm, and n is the number of fringes counted, data were collected for d and n. The data had a very high linear correlation (r = 99.99%, r 2 = 99.98%), and the regression line wasd = 167.9 + 313.6n (whered is predicted distance in nm). Since the slope of the regression line represents d/n, this regression line calculates that the wavelength of this light is 2 × 313.6nm = 627.2nm. Comparing the calculated value, 627.2nm, with the actual value, 632.8nm, yields a very small 0.88% percent error, due mainly to human handling of the precise interferometer. The high correlation between d and n deduces that d and n are in constant proportion, described by the equation λ = 2d/n, and that the wavelength of light produced by this laser is 632.8nm.
In 1897, J.J. Thomson became the first man to observe the charge to mass ratio using the apparatu... more In 1897, J.J. Thomson became the first man to observe the charge to mass ratio using the apparatus he designed, which is the same apparatus used in this experiment. Using a large magnetic field (created from the Helmholtz Coils), a current can be created from cathode rays (beams of electrons), which can be measured to find out the charge to mass ratio of the electron itself. Using classical variables, since mv 2 2
In 1644, Pietro Mengoli questioned the mathematical society about the sum of the reciprocals of t... more In 1644, Pietro Mengoli questioned the mathematical society about the sum of the reciprocals of the perfect square numbers. Thanks to Leonhard Euler in 1735, we now know that ∞ n=1
The relationship between math's two biggest constants, e and π, has been investigated by all of t... more The relationship between math's two biggest constants, e and π, has been investigated by all of the greats of mathematics -Euler, Bernoulli, et al. -but remains a haze today. 15-year-old Cody Dianopoulos speculates that the relationship between e and π is more than just that of Euler's great equation e iπ + 1 = 0, but that e and π have, in fact, a direct relationship. Dianopoulos goes into the approximations of their relationship and how computers can be used to confirm the values.
Why spend all of that time converting to component form and adding vectors when you can find the ... more Why spend all of that time converting to component form and adding vectors when you can find the magnitude of the resultant of two vectors with this new formula?
The Collatz Conjecture, first posed in 1937 by Lothar Collatz, has finally been confirmed through... more The Collatz Conjecture, first posed in 1937 by Lothar Collatz, has finally been confirmed through a series of nested proofs by fifteen-yearold Cody T. Dianopoulos.
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