answer:1. **Identify the problem and given conditions:** We are given ( p = 1601 ) and need to prove that if [ dfrac{1}{0^2+1} + dfrac{1}{1^2+1} + cdots + dfrac{1}{(p-1)^2+1} = dfrac{m}{n}, ] where the sum is taken over terms with denominators not divisible by ( p ), then ( p mid 2m + n ). 2. **Recognize the form of ( p ):** Notice that ( p = 1601 ) is of the form ( 4k + 1 ) (since ( 1601 = 4 cdot 400 + 1 )). This is crucial because for primes of this form, (-1) is a quadratic residue modulo ( p ). 3. **Use Wilson's theorem:** By Wilson's theorem, we know that [ (p-1)! equiv -1 pmod{p}. ] We can also write [ (p-1)! = left( prod_{ell=1}^{(p-1)/2} ell right) left( prod_{ell=1}^{(p-1)/2} (p-ell) right) equiv (-1)^{frac{p-1}{2}} left( left( frac{p-1}{2} right)! right)^2 pmod{p}. ] Since ( p = 1601 ), we have ( frac{p-1}{2} = 800 ), and thus [ left( 800! right)^2 equiv -1 pmod{p}. ] This implies that ( textrm{i} = 800! ) is a square root of (-1) modulo ( p ). 4. **Identify the special elements:** Since ( p = 1601 = 40^2 + 1 ), we have ( textrm{i} = 40 ) (though the exact value is not necessary for the proof). 5. **Sum over the terms:** Denote the expression by ( E ), so [ E = sum_{ell neq pm textrm{i}} dfrac{1}{ell^2 + 1}. ] We need to show that ( 2E + 1 equiv 0 pmod{p} ). 6. **Simplify the sum:** We can rewrite ( E ) as [ E = sum_{ell neq pm textrm{i}} dfrac{1}{(ell - textrm{i})(ell + textrm{i})} = dfrac{1}{2textrm{i}} sum_{ell neq pm textrm{i}} left( dfrac{1}{ell - textrm{i}} - dfrac{1}{ell + textrm{i}} right). ] This sum telescopes to [ dfrac{1}{2textrm{i}} left( dfrac{1}{2textrm{i}} - dfrac{1}{-2textrm{i}} right) = dfrac{1}{2textrm{i}^2} = -dfrac{1}{2}. ] 7. **Conclude the proof:** Therefore, we have [ E equiv -dfrac{1}{2} pmod{p}, ] which implies [ 2E + 1 equiv 0 pmod{p}. ] Hence, ( p mid 2m + n ). (blacksquare)

answer:Here's an edited version of the passage with some suggestions for improvement: The writing developed over the past 100 years, with most of its foundational ideas emerging in the 1920s, a period of rapid change in the country. During this time, immigrants from the East Coast flocked to cities like Los Angeles and San Francisco in search of jobs, bringing a level of diversity the West Coast had never seen before. However, this influx of new people also created tensions between the immigrants and native-born Americans who had lived in the area for generations. I made a few changes to enhance clarity, grammar, and flow: - Changed was developed to developed to make the sentence more concise and active. - Added foundational to ideas to provide more context and specificity. - Changed brings to bringing to correct the verb tense and maintain consistency. - Changed never been seen to had never seen to correct the verb tense. - Changed with all these new people coming in to this influx of new people to make the sentence more concise and formal. - Changed who have been living to who had lived to correct the verb tense and maintain consistency. - Changed for generation to for generations to correct the plural form. These changes should improve the overall clarity and readability of the passage.

answer:The changes I suggested contribute to a shift from informal to formal writing in several ways: 1. **Verb choice and tense correction**: Using the correct verb tense and choosing more formal verbs, such as emerging instead of were created and bringing instead of brings, contributes to a more formal tone. 2. **Word choice and specificity**: Adding words like foundational to provide more context and specificity adds a level of formality to the writing. Formal writing tends to be more precise and detailed. 3. **Sentence structure**: Reorganizing the sentences to be more concise and using transitional phrases like During this time and However creates a more formal and cohesive narrative flow. 4. **Avoiding colloquial expressions**: Phrases like all these new people coming in are more common in informal writing. Replacing them with more formal expressions like this influx of new people contributes to a formal tone. 5. **Precision in plural forms**: Using the correct plural form generations instead of generation adds to the overall formality of the writing. However, it's worth noting that the shift from informal to formal writing is a matter of degree. The original passage was not extremely informal, and the changes I suggested aimed to refine the writing rather than transform it entirely. The revised passage is still accessible and readable, but it presents the information in a more polished and formal manner, suitable for academic or professional contexts.

answer:The revised passage may have lost some of the original's sense of historical impact and drama. The original passage had a certain immediacy and simplicity that conveyed the significance of the event, particularly in the phrase brings a diversity that never been seen on the West Coast before. While the revised passage is more polished and grammatically correct, it may have become slightly more subdued in its tone. In particular: * The original passage used the phrase rapid change to describe the 1920s, which conveyed a sense of urgency and upheaval. The revised passage still mentions rapid change, but it feels slightly more abstract. * The original passage used the phrase never been seen to emphasize the unprecedented nature of the diversity brought by the immigrants. The revised passage uses the phrase had never seen, which is grammatically correct but slightly less emphatic. * The revised passage breaks up the original's longer sentences into shorter, more manageable ones, which can make the text feel slightly more measured and less dramatic. To restore the original's sense of historical impact, you could consider adding more vivid language or imagery to the revised passage, or rephrasing certain sentences to convey a greater sense of significance or urgency. For example: * Instead of saying a period of rapid change, you could say a time of seismic upheaval or an era of profound transformation. * Instead of saying a level of diversity the West Coast had never seen before, you could say a wave of diversity that reshaped the West Coast's cultural landscape or an unprecedented influx of new cultures and ideas. By adding more evocative language and imagery, you can help restore the passage's sense of historical impact and drama, while still maintaining its grammatical correctness and clarity.