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Terence Tao@What's new
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Terence Tao has recently uploaded a paper to the arXiv titled "Decomposing a factorial into large factors." The paper explores a mathematical quantity, denoted as t(N), which represents the largest value such that N! can be factorized into t(N) factors, with each factor being at least N. This concept, initially introduced by Erdös, delves into how equitably a factorial can be split into its constituent factors.
Erdös initially conjectured that an upper bound on t(N) was asymptotically sharp, implying that factorials could be split into factors of nearly uniform size for large N. However, a purported proof by Erdös, Selfridge, and Straus was lost, leading to the assertion becoming a conjecture. The paper establishes bounds on t(N), recovering a previously lost result. Further conjectures were made by Guy and Selfridge, exploring whether relationships held true for all values of N. On March 30th, mathematical enthusiasts celebrated facts related to the number 89. Eighty-nine is a Fibonacci prime, and patterns emerge when finding it's reciprocal. Also, the number 89 can be obtained by a summation of the first 5 integers to the power of the first 5 Fibonacci numbers. 89 is also related to Armstrong numbers, which are numbers that are the sum of their digits raised to the number of digits in the number.
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@tracyrenee61.medium.com
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Recent discussions have highlighted the importance of several key concepts in probability and statistics, crucial for data science and research. Descriptive measures of association, statistical tools used to quantify the strength and direction of relationships between variables are essential for understanding how changes in one variable impact others. Common measures include Pearson’s correlation coefficient and Chi-squared tests, allowing for the identification of associations between different datasets. This understanding helps in making informed decisions by analyzing the connection between different factors.
Additionally, hypothesis testing, a critical process used to make data-driven decisions, was explored. It determines if observations from data occur by chance or if there is a significant reason. Hypothesis testing involves setting a null hypothesis and an alternative hypothesis then the use of the P-value to measure the evidence for rejecting the null hypothesis. Furthermore, Monte Carlo simulations were presented as a valuable tool for estimating probabilities in scenarios where analytical solutions are complex, such as determining the probability of medians in random number sets. These methods are indispensable for anyone who works with data and needs to make inferences and predictions.
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