This page is under development. Comments are welcome, but please load any comments in the comments section at the bottom of the page. Please include your wiki MONIKER and date in your comment with the same courtesy that I will give you. Aside from your courtesy, your wiki MONIKER and date as a signature and minimal good faith of any internet post are the rules of this TCL-WIKI. Its very hard to reply reasonably without some background of the correspondent on his WIKI bio page. Thanks, gold 3/8/2024
gold Update 3/7/2024. The Twin Primes procedure is loaded on the Tcllib math::primes module. Reference Routines are listPrimePairs and listPrimeProgressions on Tcllib. Study of Twin Primes features using pseudocode. Console program outputs data as table in TCL table format and comma delimited spreadsheet. These auxiliary decks are used to proof features or subroutines. The Monopoly page from GWM seems closest in theme to what I was trying to learn. Using dice and running with random throws along a track or path like Snakes and Ladders.
gold Update 3/7/2024. The author is retired engineer on Windows 10 and no longer has proofing access to Unix machines. Unix is respected after use of so many decades in engineering, but my wings are lost. I did find a useful online IDE, jdoodle. I can paste from a text file, edit, and run an output using this online IDE.
gold Note: Some tickets for prime numbers are closed and functions available in Tcllib. Many thanks to Arjen Markus arjen AM & Andreas Kupries for the heavy lifting.
closer: arjenmarkus AM
Emailed comment from AM: I used the sample code to create two new procedures:
listPrimePairs listPrimeProgressions
The first proc listPrimePairs returns a list of pairs of primes that differ by a given number and the second proc listPrimeProgressions returns a list of arithmetic progressions of primes that differ by the given number.
This page on developing pseudocode examples and one line procedures is not a replacement for the current Tcl core and Tcllib, which is much improved since Tcl version 4, and other <faster> language constructs. math ops, Tcllib routines, and other compiled routines can reduce the cost of big-data tasks by about 1/3. The time savings of the core are not always obvious on small quantities of data, like 4 or 5 numbers. Performance of one-line programs may suffer degradation due to lengthy recursion calls, and may be limited by constraints on recursion. Dependence on math operator notation, helper procedures, math check examples, degradation due to lengthy recursion calls, and special library functions should be noted in the comment lines.
gold 3/8/2024 Draft & Check.
1. Cryptography: Primes play a crucial role in encryption algorithms such as RSA (Rivest-Shamir-Adleman), which relies on the difficulty of factoring large numbers into their prime components to secure communication online.
2. Number theory: Primes are fundamental in number theory, the branch of mathematics that studies the properties and relationships of numbers. Studying prime numbers helps to understand the distribution of primes, prime factorization, and other important concepts in mathematics.
3. Primality testing: Algorithms have been developed to efficiently determine whether a given number is prime. These algorithms are essential in various applications, such as programming languages, cryptography, and computer science.
4. Random number generation: Primes are often used in generating random numbers, especially in cryptographic applications where randomness is crucial for security. The use of prime numbers in random number generation ensures a higher level of unpredictability and security.
5. Error detection and correction: Primes are used in error detection and correction algorithms, such as the Reed-Solomon codes, which are commonly used in data storage and transmission systems. By leveraging the properties of prime numbers, these algorithms can detect and correct errors in transmitted data.
Twin primes, which are prime numbers that have a difference of 2 (e.g., 3 and 5, 11 and 13), have applications in number theory, cryptography, and computer science.
1. In number theory, the study of twin primes helps researchers understand the distribution of prime numbers.
2. In cryptography, twin primes are used in certain encryption algorithms.
3. In computer science, twin primes can be used in algorithms for generating random numbers.
# Pseudocode # Procedure: listPrimePairs # Note dependency here on Tcllib proc isprime # Parameters: # lower - lower limit for the interval # upper - upper limit for the interval # step - difference between successive primes (default: 2) # Return: # list of pairs of primes differing the given step # Procedure is_prime to check if a number is prime proc is_prime(n): if n < 2: return False for i from 2 to sqrt(n): if n % i == 0: return False return True # Procedure listPrimePairs procedure listPrimePairs(lower, upper, step=2): # Validation card if upper <= lower: return error: "The upper limit must be larger than the lower limit" if step <= 0: return error: "The step must be at least 1" # Initialize output variable here, list in TCL output = [] # for next loop in TCL for i in range(lower, upper+1): next = i + step if isPrime(i) and isPrime(next): output.append([i, next]) return output End
Pseudocode generates a list of prime number pairs that differ by the given step within the specified interval. The pseudocode uses a similar logic to the original Tcllib script, with some minor modifications to match the pseudocode syntax.
# list from copy of [Tcllib] deck on jdoodle. brun constant over selected range = series 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 = 0.7698738952 puts " list of twin primes = [ listPrimePairs 3 100 2 ] " list of twin primes = {3 5} {5 7} {11 13} {17 19} {29 31} {41 43} {59 61} {71 73} brun constant over selected range 0.5333333333333333 brun constant over selected range 0.5333333333333333 brun constant over selected range 1.330990365719087 brun constant over selected range 1.330990365719087 puts " list of twin primes = [ listPrimePairs 3 500 2 ] " bruin constant over selected range 1.486060792020912 brun very slow closer and not much core allowance on jdoodle monograph = B2(p = 10**16) = 1.902160583104 series converges extremely slowly.
Reference Testcases
Test Cases P -> Partial Sum of series for Brun Constant 2 -> 0 6 -> 0.5333333333333333 10 -> 0.8761904761904762 13 -> 0.8761904761904762 100 -> 1.3309903657190867 620 -> 1.4999706034568274 100000 -> 1.67279958482774 p B2(p) 10E2 1.330990365719 10E4 1.616893557432 10E6 1.710776930804 10E8 1.758815621067 10E10 1.787478502719 10E12 1.806592419175 10E14 1.820244968130 10E15 1.825706013240 10E16 1.830484424658 different more converging formula? Ref = Introduction to twin primes and Brun’s constant computation Pascal Sebah and Xavier Gourdon 10^5: 1.90216329186 ( 6.1 s) 10^6: 1.90191335333 ( 63.1 s) 10^7: 1.90218826322 ( 759.6 s) 2*10^7: 1.90217962170 (1692.3 s) B2(p = 10**16) => 1.902160583104
The first 60 prime gaps are:
1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, ... (sequence A001223 in the OEIS).
The average gap between primes increases as the natural logarithm of these primes, and therefore the ratio of the prime gap to the primes involved decreases (and is asymptotically zero). This is a consequence of the prime number theorem. --- Wikipedia
Taking averages between prime pairs, the positions of 2. what expected?
mean 2, 4, 6, 6, 2, mean 2, 4, 14, 4, 6, 2, mean 2, 10, 6, 6, 6, 2 puts " math_mean = [math_mean 2 4 6 6 2 ]" # 4 puts " math_mean = [math_mean 2 4 14 4 6, 2 ]" # 5 puts " math_mean = [math_mean 2 10 6 6 6, 2 ]" # 5
Omitting outside links here, too transitory on internet.
*The Twin Primes constant is defined as
*
Twin Primes Follow up with Pseudocode Slow Series Convergence for Brun Constant
gold Off the cuff, that double exponential curve fit seems to be tracking fairly well.
Credit to Wikipedia-(Kiwi128) Bruns constant in lower primes field. The convergence to B2. Each dot represents the effect of an additional pair of twin primes. While the exact value of B2 is unknown, it is thought to be around 1.9 (red line). Calculations have shown it to be greater than 1.83 (blue line)
Number theory - 2018-08-08, chat.stackexchange.com
Please include your wiki MONIKER and date in your comment with the same courtesy that I will give you. Thanks, gold 12Aug2020
test edit
Functional and Imperative Programming Functional programming computes an expression, while imperative programming is a sequence of instructions modifying memory. The former relies on an automatic garbage collector, while the latter requires explicit memory allocation and deallocation. Memory and Execution Control Imperative programming provides greater control over execution and memory representation, but can be less efficient. Functional programming offers higher abstraction and execution safety, with stricter typing and automatic storage reclamation. Historical Perspectives Historically, functional programming was associated with symbolic applications, and imperative programming with numerical applications. However, advances in compiling and garbage collection have improved efficiency and execution safety for both paradigms. The Appeal of Tool Control Language TCL TCL emphasizes that efficiency need not preempt assurance, as long as efficiency remains reasonably good. This approach is gaining popularity among software producers.
gold prompt? Improve, elaborate, explain steps, and explain definitions in numbered steps with numbered steps with details, dates, and elaboration in regard to computer applications, and Tool Control Language TCL ; use titles on each paragraph consisting of 1,2,3 or 4 words? Restrict each paragraph to 30 words or less?
gold prompt? improve, condense, elaborate, explain steps, and explain definitions in numbered steps with numbered steps with details, dates, and elaboration in regard to computer applications, and Tool Control Language TCL ; use titles on each paragraph consisting of 1,2,3 or 4 words? Restrict each paragraph to 30 words or less? Restrict text to third person and impersonal discussion text, substitute one for you, substitute ones for your, use one for he, use ones for his?
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gold Prompt: improve, condense, elaborate, explain steps, and explain definitions in numbered steps with numbered steps with details, dates, and elaboration in regard to computer applications, and Tool Control Language TCL ; use titles on each paragraph consisting of 1,2,3 or 4 words? Omit blank lines? Restrict each paragraph to 30 words or less? Restrict text to third person and impersonal discussion text, substitute one for you, substitute ones for your, use one for he, use ones for his?
Note. What about all negative values or mixed positive/negative values. Under definitions of arithmetic mean? Under math check. The procedure calculates the mean by summing all values and then dividing the sum by the number of values, regardless of the signs of the values. The arithmetic mean, also known as the average, is calculated by adding up all the values and then dividing the sum by the number of values. This definition applies to all types of values, including negative or mixed positive/negative values. The pseudocode accurately follows this definition by summing all values, regardless of their signs, and then dividing the sum by the number of values.
When calculating the arithmetic mean of a set of values, whether they are all negative, all positive, or a mix of positive and negative values, the process remains the same. The arithmetic mean is calculated by summing all the values and then dividing by the total number of values.
For example, if you have a set of negative values such as (-1, -2, -3), the arithmetic mean would be calculated by adding -1, -2, and -3 together to get -6, and then dividing by 3 (the total number of values) to get an arithmetic mean of -2.
Similarly, if you have a mix of positive and negative values (e.g., 1, -2, 3), you would sum the values (1 + (-2) + 3 = 2) and then divide by 3 to get an arithmetic mean of approximately 0.67.
The arithmetic mean calculation is not affected by the sign of the values, as it is a measure of central tendency that considers the magnitude of the values rather than their sign. ---
# pseudocode # example to illustrate this process: # Function to check if a number is prime function is_prime(n): if n < 2: return False for i from 2 to sqrt(n): if n % i == 0: return False return True # Function to find twin prime pairs and calculate the gap function find_twin_primes(): twin_primes = [] for i from 2 to MAX_NUMBER: if is_prime(i) and is_prime(i + 2): twin_primes.append((i, i + 2)) prime_gaps = [] for pair in twin_primes: gap = pair[1] - pair[0] prime_gaps.append(gap) return prime_gaps # Function to calculate the average of prime gaps function calculate_average(gaps): total = 0 for gap in gaps: total += gap average = total / len(gaps) return average # Main program twin_prime_gaps = find_twin_primes() average_gap = calculate_average(twin_prime_gaps) print("Average gap between twin primes: ", average_gap)
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