Chronos notelife نرم افزاری برای نگهداری و دسته بندی یادداشت های شماست. نرم افزاری سبک با رابط کاربری ساده است / notelife is a premium note manager designed for the rest of us. While it goes beyond the basics, it isnt hard to use. A good storage system for IDEAS is essential. I have been able to sync with my laptop pretty easily by setting up a computer-to-computer network on my laptop, turning on wireless on my iPhone and joining the network, and then opening up NoteLife and syncking.
I was just solving some congruence problems with prime modulo. Such as this one:
[tex]x^2equiv 59(mod 11)[/tex]
I was using Quadratic Residues to determine whether or not my equations had solutions. For example, in the case above, I would write out the quadratic residues of 11:
[tex]QR_{11} = {1,3,4,5,9}[/tex]
Here I noted that if my modulo was prime then the number of quadratic residues is always less than half the prime. So in this case, 11/2 = 5.5 and there are 5 QR's.
Then I would use the Legendre symbol:
[tex](59/11) = (4/11) = 1 quadquadmbox{since 4 is a quadratic residue of 11}[/tex]
Which tells me that my equation has a solution. But notice that I needed to calculate all the quadratic residues first.
So then I came across the Quadratic Reciprocity Law which said that if p and q are primes then
[tex](p/q) = -(q/p)[/tex]
if [itex]p,qequiv 3(mod 4)[/itex], otherwise [itex](p/q) = (q/p)[/itex].
So, this tells me that solving an equation such as [itex]x^2equiv p(mod q)[/itex] is exactly the same as solving [itex]x^2 equiv q(mod p)[/itex], UNLESS p or q is congruent to 3 modulo 4.
EX1. In modulo 5, we have 1 and 4 as quadratic residues. Then (5/97) = (97/5) because neither 5 nor 97 are congruent to 3 modulo 4. So then (5/97) = (97/5) = (2/5) = -1 since 2 is not a QR of 5.
EX2. (85/97) = (5/97)(17/97) {by a corollary of the QRL} = (97/5)(97/17) {since neither 5 nor 17 are congruent to 3 modulo 4} = (2/5)(12/17) {simplifying in mod 17} = (-1)(-1) = 1. Since the QR's of 17 are 1,2,4,8,9,13,15,16, and 12 is not a QR of 17.
My question is: why is what I've written in bold true? Why aren't those two equations equivalent if p or q is congruent to 3 modulo 4? What is so special about 3 modulo 4?
[tex]x^2equiv 59(mod 11)[/tex]
I was using Quadratic Residues to determine whether or not my equations had solutions. For example, in the case above, I would write out the quadratic residues of 11:
[tex]QR_{11} = {1,3,4,5,9}[/tex]
Here I noted that if my modulo was prime then the number of quadratic residues is always less than half the prime. So in this case, 11/2 = 5.5 and there are 5 QR's.
Then I would use the Legendre symbol:
[tex](59/11) = (4/11) = 1 quadquadmbox{since 4 is a quadratic residue of 11}[/tex]
Which tells me that my equation has a solution. But notice that I needed to calculate all the quadratic residues first.
So then I came across the Quadratic Reciprocity Law which said that if p and q are primes then
[tex](p/q) = -(q/p)[/tex]
if [itex]p,qequiv 3(mod 4)[/itex], otherwise [itex](p/q) = (q/p)[/itex].
So, this tells me that solving an equation such as [itex]x^2equiv p(mod q)[/itex] is exactly the same as solving [itex]x^2 equiv q(mod p)[/itex], UNLESS p or q is congruent to 3 modulo 4.
EX1. In modulo 5, we have 1 and 4 as quadratic residues. Then (5/97) = (97/5) because neither 5 nor 97 are congruent to 3 modulo 4. So then (5/97) = (97/5) = (2/5) = -1 since 2 is not a QR of 5.
EX2. (85/97) = (5/97)(17/97) {by a corollary of the QRL} = (97/5)(97/17) {since neither 5 nor 17 are congruent to 3 modulo 4} = (2/5)(12/17) {simplifying in mod 17} = (-1)(-1) = 1. Since the QR's of 17 are 1,2,4,8,9,13,15,16, and 12 is not a QR of 17.
My question is: why is what I've written in bold true? Why aren't those two equations equivalent if p or q is congruent to 3 modulo 4? What is so special about 3 modulo 4?
Nate Life
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• 1.0.3 The “Unfiled” folder name can now be edited
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• 1.0.3 Fixed some problems displaying emojis in the Note List
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• 1.0.2 Added compatibility with iOS 11
• 1.0.2 Added 3D Touch Quick Action to app icon: New Note, Photo Library, Take Photo and Search
• 1.0.2 Photos can now be added to the Media Gallery (above the note) or inserted in the note text (Settings > General)
• 1.0.2 When a photo is inserted in the note text, a thumbnail now appears in the Note List
• 1.0.2 A search now highlights the search terms in the Note List
• 1.0.2 Fixed problem dismissing the Tags popover when running on an iPad
• 1.0.2 Redesigned the Settings view to make it easier to use
• 1.0.2 Fixed some problems with the note text view
• 1.0.2 Fixed problem with keyboard sometimes not appearing when tapping an note’s text
• 1.0.2 Images attached to a note are now printed at full-size instead of thumbnail size
• 1.0.2 Photos embedded in text resize automatically
• 1.0.2 Faster iCloud sync
• 1.0.2 Added Debug Log feature
• 1.0.2 Fixed problem where the app occasionally wouldn’t recognize a subscription
• 1.0.1 Added ability to bold the first line of new notes to signify a pseudo-title
• 1.0.1 Added setting to control automatic automatic bolding of first line of new notes
• 1.0.1 Added ability to sort notes by “title' (first line of note since notes don’t have an actual title)
• 1.0.5 iCloud syncing improvements (including shorter timeouts for poor connections)
• 1.0.4 iPhone X compatibility including Face ID support
• 1.0.4 Combined font and tag controls into single keyboard toolbar to make it modeless
• 1.0.4 Fixed problem updating font alignment controls when no text is selected
• 1.0.4 Fixed handling of mute switch on device to match behavior of Apple’s apps
• 1.0.4 Fixed a problem where moving a note to a new folder couldn’t be cancelled
• 1.0.3 Redesigned the Folder List and Note List views for iOS 11
• 1.0.3 Important changes to improve iCloud syncing
• 1.0.3 The Note List includes an integrated search field—pull the Note List down to access
• 1.0.3 The number of notes appears in the Note List’s search field
• 1.0.3 If the Note List is being filtered, the matching number of notes appears in the navigation bar
• 1.0.3 The Note Info sidebar shows the word, character and paragraph counts for the note text
• 1.0.3 A sync status bar appears at the bottom of the Folder List during a sync
• 1.0.3 Settings includes sync status information and controls for manually triggering a sync, etc.
• 1.0.3 The app can no longer go to sleep while a large sync is running
• 1.0.3 The “Unfiled” folder name can now be edited
• 1.0.3 Added “I forgot my password” button to the Privacy settings
• 1.0.3 When a note was being moved and the Cancel button was tapped, the Cancel button now immediately disappears
• 1.0.3 Fixed some problems displaying emojis in the Note List
• 1.0.3 Fixed a problem where the Done button in the photo viewer was obscured if there were a lot of photos attached
• 1.0.2 Added compatibility with iOS 11
• 1.0.2 Added 3D Touch Quick Action to app icon: New Note, Photo Library, Take Photo and Search
• 1.0.2 Photos can now be added to the Media Gallery (above the note) or inserted in the note text (Settings > General)
• 1.0.2 When a photo is inserted in the note text, a thumbnail now appears in the Note List
• 1.0.2 A search now highlights the search terms in the Note List
• 1.0.2 Fixed problem dismissing the Tags popover when running on an iPad
• 1.0.2 Redesigned the Settings view to make it easier to use
• 1.0.2 Fixed some problems with the note text view
• 1.0.2 Fixed problem with keyboard sometimes not appearing when tapping an note’s text
• 1.0.2 Images attached to a note are now printed at full-size instead of thumbnail size
• 1.0.2 Photos embedded in text resize automatically
• 1.0.2 Faster iCloud sync
• 1.0.2 Added Debug Log feature
• 1.0.2 Fixed problem where the app occasionally wouldn’t recognize a subscription
• 1.0.1 Added ability to bold the first line of new notes to signify a pseudo-title
• 1.0.1 Added setting to control automatic automatic bolding of first line of new notes
• 1.0.1 Added ability to sort notes by “title' (first line of note since notes don’t have an actual title)