User:Eliora/Sandbox: Difference between revisions

Line 49:

This made the Fermat-prime result one of the principal links in the chain of otherwise unrelated discoveries that later characterized the Day of Mathematics.

The discovery also had an immediate consequence in classical Euclidean geometry. By the Gauss–Wantzel theorem, a regular <math>n</math>-gon is constructible using only an unmarked straightedge and compass when math>n</math> is the product of a power of two and distinct Fermat primes. The primality of <math>F_M</math> therefore established the existence of an entirely new (albeit an unimaginably large) family of constructible regular polygons involving math>F_M</math> as a factor, including the regular math>F_M</math>-gon itself. It was the first expansion of the known set of Fermat-prime factors available for constructible polygons since Gauss's 1796 discovery of the constructibility of the regular 17-gon. Although an explicit straightedge-and-compass construction of a polygon with the number of sides of math>F_M</math> magnitude was of no practical geometric significance, the result attracted considerable attention as an unusual instance in which a computational discovery involving a number of extraordinary size immediately enlarged a classification originating in ancient Greek geometry.

The discovery also had an immediate consequence in classical Euclidean geometry. By the Gauss–Wantzel theorem, a regular <math>n</math>-gon is constructible using only an unmarked straightedge and compass when <math>n</math> is the product of a power of two and distinct Fermat primes. The primality of <math>F_M</math> therefore established the existence of an entirely new (albeit an unimaginably large) family of constructible regular polygons involving <math>F_M</math> as a factor, including the regular <math>F_M</math>-gon itself. It was the first expansion of the known set of Fermat-prime factors available for constructible polygons since Gauss's 1796 discovery of the constructibility of the regular 17-gon. Although an explicit straightedge-and-compass construction of a polygon with the number of sides of <math>F_M</math> magnitude was of no practical geometric significance, the result attracted considerable attention as an unusual instance in which a computational discovery involving a number of extraordinary size immediately enlarged a classification originating in ancient Greek geometry.

The exact ordinal position of <math>N_2</math> among the odd perfect numbers (or perfect numbers overall, for that matter) remains unknown and is considered beyond current methods of determination. Preliminary work on their distribution suggested that the magnitude of the k-th odd perfect number grows at least approximately as <math>c^{c^{k}}</math>, while proposed upper estimates conjecture growth on a tetrational scale, <math>c \uparrow \uparrow k</math>, for a constant <math>c</math> whose value remains undetermined. The extreme sparsity implied by these estimates has also led to the conjecture that, if infinitely many odd perfect numbers exist, the sum of their reciprocals would be a {{W|Liouville number}}.

@@ Line 49: / Line 49: @@
 This made the Fermat-prime result one of the principal links in the chain of otherwise unrelated discoveries that later characterized the Day of Mathematics.
-The discovery also had an immediate consequence in classical Euclidean geometry. By the Gauss–Wantzel theorem, a regular <math>n</math>-gon is constructible using only an unmarked straightedge and compass when math>n</math> is the product of a power of two and distinct Fermat primes. The primality of <math>F_M</math> therefore established the existence of an entirely new (albeit an unimaginably large) family of constructible regular polygons involving math>F_M</math> as a factor, including the regular math>F_M</math>-gon itself. It was the first expansion of the known set of Fermat-prime factors available for constructible polygons since Gauss's 1796 discovery of the constructibility of the regular 17-gon. Although an explicit straightedge-and-compass construction of a polygon with the number of sides of math>F_M</math> magnitude was of no practical geometric significance, the result attracted considerable attention as an unusual instance in which a computational discovery involving a number of extraordinary size immediately enlarged a classification originating in ancient Greek geometry.
+The discovery also had an immediate consequence in classical Euclidean geometry. By the Gauss–Wantzel theorem, a regular <math>n</math>-gon is constructible using only an unmarked straightedge and compass when <math>n</math> is the product of a power of two and distinct Fermat primes. The primality of <math>F_M</math> therefore established the existence of an entirely new (albeit an unimaginably large) family of constructible regular polygons involving <math>F_M</math> as a factor, including the regular <math>F_M</math>-gon itself. It was the first expansion of the known set of Fermat-prime factors available for constructible polygons since Gauss's 1796 discovery of the constructibility of the regular 17-gon. Although an explicit straightedge-and-compass construction of a polygon with the number of sides of <math>F_M</math> magnitude was of no practical geometric significance, the result attracted considerable attention as an unusual instance in which a computational discovery involving a number of extraordinary size immediately enlarged a classification originating in ancient Greek geometry.
 The exact ordinal position of <math>N_2</math> among the odd perfect numbers (or perfect numbers overall, for that matter) remains unknown and is considered beyond current methods of determination. Preliminary work on their distribution suggested that the magnitude of the k-th odd perfect number grows at least approximately as <math>c^{c^{k}}</math>, while proposed upper estimates conjecture growth on a tetrational scale, <math>c \uparrow \uparrow k</math>, for a constant <math>c</math> whose value remains undetermined. The extreme sparsity implied by these estimates has also led to the conjecture that, if infinitely many odd perfect numbers exist, the sum of their reciprocals would be a {{W|Liouville number}}.