Solving the Odd Perfect Number Problem: Some New
... σ(pi αi ) = M . (Note that, since M is odd, αi must be even.) Applying the σ function to both sides of the last equation, we get σ(σ(pi αi )) = σ(M ) = 2pi αi , which means that pi αi is an odd superperfect number. But Kanold [4] showed that odd superperfect numbers must be perfect squares (no contr ...
... σ(pi αi ) = M . (Note that, since M is odd, αi must be even.) Applying the σ function to both sides of the last equation, we get σ(σ(pi αi )) = σ(M ) = 2pi αi , which means that pi αi is an odd superperfect number. But Kanold [4] showed that odd superperfect numbers must be perfect squares (no contr ...
Fermat*s Little Theorem (2/24)
... 341 is called a 2-pseudoprime (i.e., “false prime with respect to base 2”). There are in fact infinitely many. The smallest 3-pseudoprime is 91. Etc. Really disturbing: A Carmichael number is a k-pseudoprime for every base k to which it is relatively prime. 561 is the smallest. There are infinitely ...
... 341 is called a 2-pseudoprime (i.e., “false prime with respect to base 2”). There are in fact infinitely many. The smallest 3-pseudoprime is 91. Etc. Really disturbing: A Carmichael number is a k-pseudoprime for every base k to which it is relatively prime. 561 is the smallest. There are infinitely ...
Analytic Functions
... general functions of a complex variable. Once we have proved results to determine whether or not a function is analytic, we shall then consider generalizations of some of the more common single variable functions which are not polynomials - namely trigonometric functions and exponential functions. 1 ...
... general functions of a complex variable. Once we have proved results to determine whether or not a function is analytic, we shall then consider generalizations of some of the more common single variable functions which are not polynomials - namely trigonometric functions and exponential functions. 1 ...
Combinatorial Mathematics Notes
... Remark. Sum Principle: If a finite set A is partitioned into sets B1 , . . . , Bk , then A k i 1 Bi . Product Principle: If the elements of set A are built via successive choices, where the number of options for the ith choice is independent of the outcomes of the earlier choices, then A is the pro ...
... Remark. Sum Principle: If a finite set A is partitioned into sets B1 , . . . , Bk , then A k i 1 Bi . Product Principle: If the elements of set A are built via successive choices, where the number of options for the ith choice is independent of the outcomes of the earlier choices, then A is the pro ...
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals.