March - The Euler Archive - Mathematical Association of America
... and Euler proof of Theorem 8 is almost exactly like the proof of Theorem 7, with the exponent n included. Also, the proof of Theorem 8 is correct by modern standards because all the series involved are absolutely convergent. Occasionally, someone will insist that, because Euler proved this Theorem 8 ...
... and Euler proof of Theorem 8 is almost exactly like the proof of Theorem 7, with the exponent n included. Also, the proof of Theorem 8 is correct by modern standards because all the series involved are absolutely convergent. Occasionally, someone will insist that, because Euler proved this Theorem 8 ...
the Catalan numbers
... which proves the claim for these two cases. Notice that in the second case, there are a priori n terms in ...
... which proves the claim for these two cases. Notice that in the second case, there are a priori n terms in ...
Full text
... number of fc-element independent sets in the 1 xn lattice, i.e., in a path on n vertices. Burosch suggested to consider other graphs, and some results were obtained in [3]. Answering a question of Weber, the number of independent sets in the Hasse graph of the Boolean lattice was determined asymptot ...
... number of fc-element independent sets in the 1 xn lattice, i.e., in a path on n vertices. Burosch suggested to consider other graphs, and some results were obtained in [3]. Answering a question of Weber, the number of independent sets in the Hasse graph of the Boolean lattice was determined asymptot ...
Prime Numbers
... A mathematical proof is as a carefully reasoned argument to convince a sceptical listener (often yourself) that a given statement is true. Both discovery and proof are integral parts of problem solving. When you think you have discovered that a certain statement is true, try to figure out why it is ...
... A mathematical proof is as a carefully reasoned argument to convince a sceptical listener (often yourself) that a given statement is true. Both discovery and proof are integral parts of problem solving. When you think you have discovered that a certain statement is true, try to figure out why it is ...
PERMUTATIONS WITHOUT 3-SEQUENCES 1. Introduction, The
... to this set of sequences. More recently, Kaplansky [2] and Wolfowitz [4] have enumerated permutations without rising or falling 2-sequences, that is, without 21,32, • • • ,n n — l a s w e l l a s 12, • • -,w —1 n. An addition to these results, the enumeration of permutations without 3-sequences, 123 ...
... to this set of sequences. More recently, Kaplansky [2] and Wolfowitz [4] have enumerated permutations without rising or falling 2-sequences, that is, without 21,32, • • • ,n n — l a s w e l l a s 12, • • -,w —1 n. An addition to these results, the enumeration of permutations without 3-sequences, 123 ...
Equivalents of the (Weak) Fan Theorem
... (β)i. Then for all x there exist i,n such that the nth approximation of x is between the nth approximation of (α)i and the nth approximation of (β)i: (α)i(n)00 < x(n)0 ≤ x(n)00 < (β)i(n)0 Then (by making n bigger): for all x there exist i,n such that i < n and the nth approximation of x is between t ...
... (β)i. Then for all x there exist i,n such that the nth approximation of x is between the nth approximation of (α)i and the nth approximation of (β)i: (α)i(n)00 < x(n)0 ≤ x(n)00 < (β)i(n)0 Then (by making n bigger): for all x there exist i,n such that i < n and the nth approximation of x is between t ...
Chapter 3 Proof
... is that there are not any. Resolving this question one way or another will assure you of a nice career in mathematics. Let us suppose that the conjecture is false, and that someone produces a number x that is both odd and a perfect number. The number x would then be a counterexample to the conjectur ...
... is that there are not any. Resolving this question one way or another will assure you of a nice career in mathematics. Let us suppose that the conjecture is false, and that someone produces a number x that is both odd and a perfect number. The number x would then be a counterexample to the conjectur ...
6 Prime Numbers
... Aside It is a hard problem to prove that a given large integer (say with 150 digits) is prime. But some very large primes are known, such as 257,885,161 − 1 with 17, 425, 170 digits, found on January 25th , 2013, by the Great Internet Mersenne Prime Search End of ...
... Aside It is a hard problem to prove that a given large integer (say with 150 digits) is prime. But some very large primes are known, such as 257,885,161 − 1 with 17, 425, 170 digits, found on January 25th , 2013, by the Great Internet Mersenne Prime Search End of ...
PDF file
... Hence, |M (ps )| can equal |K(ps )| − 1 only if ϕ(q) = q and 2k−1 qps−2 = 1. This can occur if and only if q = k = 1 and s = 2. Therefore, p − 1 = 2, which implies that n = 32 = 9. By inspection, we find that K(9) = {2, 5, 8}, M (9) = {2, 5}, and thus |M (9)| = |K(9)| − 1. Since M (9) = ∅, it follo ...
... Hence, |M (ps )| can equal |K(ps )| − 1 only if ϕ(q) = q and 2k−1 qps−2 = 1. This can occur if and only if q = k = 1 and s = 2. Therefore, p − 1 = 2, which implies that n = 32 = 9. By inspection, we find that K(9) = {2, 5, 8}, M (9) = {2, 5}, and thus |M (9)| = |K(9)| − 1. Since M (9) = ∅, it follo ...
Full text
... that, for any natural number a and any m, there is a prime that divides am - 1 but does not divide ak - 1 for k < m with a small number of explicitly stated exceptions. A summary of Zsigmondy's article can be found in [2, Vol. 1, p. 195]. Since the arithmetic behavior of the sequence of Fibonacci nu ...
... that, for any natural number a and any m, there is a prime that divides am - 1 but does not divide ak - 1 for k < m with a small number of explicitly stated exceptions. A summary of Zsigmondy's article can be found in [2, Vol. 1, p. 195]. Since the arithmetic behavior of the sequence of Fibonacci nu ...
HW 7. - U.I.U.C. Math
... The sum of the boxed entries is 3 + 8 + 9 + 14 = 34 for the matrix on the left, and 2 + 5 + 12 + 15 = 34, for the matrix on the right. Trying out more examples one always obtains 34 as sum of the chosen entries, suggesting that M4 = 34 is the “magic number” for the case n = 4. Your task is to prove ...
... The sum of the boxed entries is 3 + 8 + 9 + 14 = 34 for the matrix on the left, and 2 + 5 + 12 + 15 = 34, for the matrix on the right. Trying out more examples one always obtains 34 as sum of the chosen entries, suggesting that M4 = 34 is the “magic number” for the case n = 4. Your task is to prove ...
Guidelines for Solving Related-Rates Problems 1. Identify all given
... be determined. Make a sketch and label the quantities. 2. Write an equations involving the variables whose rates of change either are given or are to be determined. 3. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time t. 4. Substitute into the resulting e ...
... be determined. Make a sketch and label the quantities. 2. Write an equations involving the variables whose rates of change either are given or are to be determined. 3. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time t. 4. Substitute into the resulting e ...
Dr. Z`s Math151 Handout #4.3 [The Mean Value Theorem and
... 1. Verify that the function f (x) is continuous on [a, b] and differentiable on (a, b). For polynomials, exponential, and the ‘nice’ trig functions (sin x and cos x) this is always true. You only have to watch out for rational functions, where the bottom might vanish in the interval, and any express ...
... 1. Verify that the function f (x) is continuous on [a, b] and differentiable on (a, b). For polynomials, exponential, and the ‘nice’ trig functions (sin x and cos x) this is always true. You only have to watch out for rational functions, where the bottom might vanish in the interval, and any express ...
Chapter 7 - pantherFILE
... board Decide who goes first. In turn, write one of your numbers on the game board. A number may be used only once. Add each row and columns. Write the sums. If more sums are +, + wins. After you have played a few games, write addition patterns you see. ...
... board Decide who goes first. In turn, write one of your numbers on the game board. A number may be used only once. Add each row and columns. Write the sums. If more sums are +, + wins. After you have played a few games, write addition patterns you see. ...
partitions with equal products (ii) 76 • 28 • 27 = 72 • 38 • 21 = 57 • 56
... these partitions are mutually disjoint, i.e., no integer occurs in more than one of them. Of some additional interest is a lemma stating that a certain class of elliptic curves has positive rank over Q. ...
... these partitions are mutually disjoint, i.e., no integer occurs in more than one of them. Of some additional interest is a lemma stating that a certain class of elliptic curves has positive rank over Q. ...
Full text
... Over 20,000 problems from 38 journals and 21 contests are referenced by the site, which was developed by Stanley Rabinowitz's MathPro Press. Ample hosting space for the site was generously provided by the Department of Mathematics and Statistics at the University of MissouriRolla, through Leon M. Ha ...
... Over 20,000 problems from 38 journals and 21 contests are referenced by the site, which was developed by Stanley Rabinowitz's MathPro Press. Ample hosting space for the site was generously provided by the Department of Mathematics and Statistics at the University of MissouriRolla, through Leon M. Ha ...