An interesting example of a theorem of the theory of numbers that has

Question

An interesting example of a theorem of the theory of numbers that has been neither proven nor disproven is the so-called Goldbach conjecture, proposed in the year 1742. It states that each even number can be represented as the sum of two primes. One can easily find that it is true as applied to some simple examples, thus: 12=7+5, 24=17+7, and 32=29+3. But, in spite of the immense amount of work done in this line, mathematicians have never been able either to give conclusive proof of the infallibility of this statement or to find an example that would disprove it.

As recently as 1931, a Russian mathematician, Schnirelman, succeeded in taking the first constructive step towards securing the desired proof. He was able to show that each even number is the sum of not more than 300,000 primes. Still more recently the gap between Schnirelman’s “sum of three hundred thousand primes” and the desired “sum of two primes” was considerably narrowed by another Russian mathematician, Vinogradoff, who was able to reduce it to “the sum of four primes”. But the  last two steps  from Vinogradoff’s four to Goldbach’s two primes seem to be the toughest of all and nobody can tell whether another few years or another few centuries will be required to prove or disprove this difficult proposition.   1 
 1. Which of the following most accurately states the main idea in the passage?

A. Goldbach conjecture is a fundamental theorem of the theory of numbers.
B. Goldbach was a mathematical genius unparalleled since his time.
C. Goldbach conjecture can be verified with simple examples.
D. Goldbach conjecture is wanting a mathematical verification.
E. A significant portion of Goldbach conjecture has been proven.

2 
 2. Which of the following can be most directly inferred from the usage of the term "last two steps" (see highlighted text)?

A. The author conjectures that the yet-unproven part of Goldbach conjecture can be proven in two steps.
B. It is a term figuratively used to suggest that the seemingly small number of remaining steps are likely the toughest.
C. The proof of "sum of six primes" was two steps before the proof of "sum of four primes".
D. There were three hundred thousand steps in the proof of Goldbach conjecture only two of which are yet to be completed.
E. The number of steps pending in a proof is inversely proportional to the difficulty of those steps.

3 
 3. Which of the following is most consistent with the world of mathematical theorems as indicated by the passage?

A. All mathematical theorems apart from Goldbach conjecture are proven.
B. Mathematical proofs that can be verified with simple examples are difficult to prove.
C. Even if a theorem can be easily verified, the theorem is expected to be proven.
D. New mathematical explanations cannot offer proof to mathematical theorems previously verified to be true.
E. There is significant uncertainty regarding the proof of all difficult mathematical propositions.

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agrasan · Answer

1. Which of the following most accurately states the main idea in the passage?

A. Goldbach conjecture is a fundamental theorem of the theory of numbers.
 Incorrect , not the main idea, it is just an introduction of this theorem.

B. Goldbach was a mathematical genius unparalleled since his time.
 Incorrect , not the main idea which the author is talking about.

C. Goldbach conjecture can be verified with simple examples.
 Incorrect , not the main idea as it talks about only the first passage but there is more story after that.

D. Goldbach conjecture is wanting a mathematical verification.
 Correct , still after decent work by two mathematicians, verification is pending.

E. A significant portion of Goldbach conjecture has been proven.
 Incorrect , not the main idea, the passage in fact doesn't state this.

2. Which of the following can be most directly inferred from the usage of the term "last two steps" (see highlighted text)?

A. The author conjectures that the yet-unproven part of Goldbach conjecture can be proven in two steps.
 Incorrect , not stated at all.

B. It is a term figuratively used to suggest that the seemingly small number of remaining steps are likely the toughest.
 Correct , the reference line is  "But the last two steps from Vinogradoff’s four to Goldbach’s two primes seem to be the toughest of all and nobody can tell whether another few years or another few centuries will be required to prove or disprove this difficult proposition."

C. The proof of "sum of six primes" was two steps before the proof of "sum of four primes".
 Incorrect , "sum of six primes" is not stated at all.

D. There were three hundred thousand steps in the proof of Goldbach conjecture only two of which are yet to be completed.
 Incorrect , 300,000 steps were in Schnirelman's work, not Goldbach's work.

E. The number of steps pending in a proof is inversely proportional to the difficulty of those steps.
 Incorrect , we cannot infer this general relationship just based on one data point.

3. Which of the following is most consistent with the world of mathematical theorems as indicated by the passage?

A. All mathematical theorems apart from Goldbach conjecture are proven.
 Incorrect , "All' is a very big stretch based on given passage.

B. Mathematical proofs that can be verified with simple examples are difficult to prove.
 Incorrect , no comment has been specifically on  mathematical proofs , the passage is about mathematical theorems.

C. Even if a theorem can be easily verified, the theorem is expected to be proven.
 Correct , the reference line is  "One can easily find that it is true as applied to  some simple examples , thus: 12=7+5, 24=17+7, and 32=29+3. But, in spite of the immense amount of work done in this line, mathematicians have never been able either to give conclusive proof of the infallibility of this statement or to find an example that would disprove it."

D. New mathematical explanations cannot offer proof to mathematical theorems previously verified to be true.
 Incorrect , goes in opposite direction of the passage.

E. There is significant uncertainty regarding the proof of all difficult mathematical propositions.
 Incorrect , "All' is a very big stretch based on given passage.

vasu1104 · Answer

passage 1= the theorem of the theory of numbers by goldbach has been neither proven not disproven. there is significant amount of work being done behind it but nothing has been able to give proof to realiability of statement or disprove it.

passage 2 = two russian did job to prove it but they have not been able to arrive exactly at what Goldbach has arrived

Que 1.  A,B and C are easy to eliminate. E sound good but the passage is not concern with how much of his thesis has been proven. so reject it. 
 D defines exactly what passage set to depict. the first line and last line of first passage gives the idea of passage. so this is clear winner.

Que 2.  passage is basically saying that one mathematician came with some not more than 300000 prime while other reduce this to only 4 prime. but Goldbach has only 2 prime number in this theorem. so author is saying that to go from 4 to 2 is tough task to do even it seems small num and it will take years or centuries to prove it disprove it.

only B defines this meaning. 
 
Que 3. i got it from POE. A and E are too extreme for using ALL in the choice. reject.
D goes in other direction. actually the two new explanation did the job but not at level at which goldbach did so reject it.
B talks about proofs but passage talks about theorem. so reject it.
only left is C so is the ans

Sarvxdnya · Answer

Question number 3 ,and is option C 
 Explanation : THe question asked the statement whch can be easily deduced as a truth from the passage , in the provided option , C makes sense cause as per the explanation mentioned in paragraph 1,Line 5-8 it state that it can be easily verified from simple examples.

Dereno · Answer

An interesting example about a theorem that is neither proven or disproven is the Goldbach Conjecture. This theorem is proposed in 1742, which states that  even numbers can be represented as sum of two prime numbers.  While simple examples are showing positive signs for this theorem. Despite so much of efforts put in these line, the conclusive proof about the infallibility of this theorem or evidences to disprove it doesn’t exist.

So, in 1931, a mathematician named - Schnirelman, has taken constructive steps towards securing the proof needed to prove this conjecture. He was able to show even numbers are sum of not more than 300,000 prime numbers. The huge gap of 300,000 prime numbers is further narrowed down by Russian mathematician, Vinogradoff, who was able to reduce it to “the sum of four primes”.

Inspite of such valiant efforts by these two mathematicians, they can’t move towards the last two steps, from 4 primes to two primes of Goldbach is the toughest of all. None can predict the exact time frame for these outcomes. So, it’s difficult to prove or disprove the hypothesis till then.

Let’s move to the questions.

1. Which of the following most accurately states the main idea in the passage?

A. Goldbach conjecture is a fundamental theorem of the theory of numbers.

Theory of numbers is a broader concept, and this conjecture is a tip of an iceberg. Hence, Wrong.

B. Goldbach was a mathematical genius unparalleled since his time.

We are not questioning his ability or mathematical skills in this passage. The passage is entirely about a conjecture put forth in 1742 by Goldbach. Hence, wrong.

C. Goldbach conjecture can be verified with simple examples.

Yeah, it’s absolutely true that Goldbach conjecture can be verified using simple examples, but that’s not the main idea of the passage. Hence, can be eliminated.

D. Goldbach conjecture is wanting a mathematical verification.

This is the main theme of the passage, as from the initial line, this states that this conjecture is neither proven or disproven till date. And proceeds to explain the timeline of events which validates this line. Hence, Correct answer. It has been a long wait for the conjecture to be given a side to take.

E. A significant portion of Goldbach conjecture has been proven.

We know that some constructive steps have been taken by two Russian mathematicians from 1931, in this direction. But, until unless this conjecture is proven or disproven, we cannot definitely say that these contributions are significant portion or not. Moreover, sum of two primes is different from 300,000 and 4 primes. Hence, Wrong.

Option D

Lets move to the next question.

2. Which of the following can be most directly inferred from the usage of the term "last two steps" (see highlighted text)?

A. The author conjectures that the yet-unproven part of Goldbach conjecture can be proven in two steps.

This is not the inference, the author clearly states that narrowing from 300,000 primes down to 4 primes cannot be considered as a tough task. But, the last few steps are extremely tough and daunting task to get accomplished. That’s why in the last line, the author concludes the exact time frame of completion cannot be given. Hence, Wrong.

B. It is a term figuratively used to suggest that the seemingly small number of remaining steps are likely the toughest.

This is the exact inference of the word, the last two steps. This is mentioned to show its toughness and extreme time consuming nature in proving or disproving the conjecture. Hence, Correct answer.

C. The proof of "sum of six primes" was two steps before the proof of "sum of four primes".

This can be considered as a random guess or back tracked assumption, which is not true. Thus , it tries to prove taking two steps will land you in the treasure chest. Hence, Wrong.

D. There were three hundred thousand steps in the proof of Goldbach conjecture only two of which are yet to be completed.

This assumes that Goldbach conjecture is a 300,000 step process, while the Two Russians were smart enough to narrow the gap. The phrase last two steps, denotes the penultimate steps in sequence. Completely wrong, as we have failed to capture the tone and path of the author. Hence, Wrong.

E. The number of steps pending in a proof is inversely proportional to the difficulty of those steps .

This option mentions the difficulty aspect, but relates it with more steps is easier than fewer steps. Tries to bring a correlation for an event that has occured naturally. Hence, Wrong.

Option B.

3. Which of the following is most consistent with the world of mathematical theorems as indicated by the passage?

A. All mathematical theorems apart from Goldbach conjecture are proven.

This makes a blanket statement assumption about mathematical theories. The only theory which is left to be proven is the Goldbach conjecture. Completely wrong, as there is not a single mention of other theories.

B. Mathematical proofs that can be verified with simple examples are difficult to prove.

This makes a generalised claim, that when it’s simpler to verify it using simpler examples, it’s difficult to prove. What has happened for Goldbach conjecture cannot be taken as only outcome for all other theories. And, the passage is about the Specifc hypothesis in debate and not about other theories.

C. Even if a theorem can be easily verified, the theorem is expected to be proven.

This is a correct answer. As any conjecture , how simple can be the explanation to verify it, but until proved it remains as a hypothesis or conjecture and cannot claim the status of Theorem.

D. New mathematical explanations cannot offer proof to mathematical theorems previously verified to be true.

This brings an entire new context to paint the concept with a different perspective, which is illogical and wrong.

E. There is significant uncertainty regarding the proof of all difficult mathematical propositions.

This goes into generalising things based on a single conjecture, and that too without any evidence or proof supporting this claim. Hence, wrong.

Option C

ExpertsGlobal5 · Answer

Mind-map

To indicate that Goldbach conjecture is neither proven nor disproven (Paragraph 1) 
 To mention two widely scattered attempts towards securing the conjecture’s as-yet-missing proof (Paragraph 2)

= = = = == = = = =

1. Which of the following most accurately states the main idea in the passage?

A. Goldbach conjecture is a fundamental theorem of the theory of numbers.
B. Goldbach was a mathematical genius unparalleled since his time.
C. Goldbach conjecture can be verified with simple examples.
D. Goldbach conjecture is wanting a mathematical verification.
E. A significant portion of Goldbach conjecture has been proven.

Question Type: Inference

Explanation:

As the mind-map suggests, the first paragraph indicates that Goldbach conjecture is neither proven nor disproven and the second paragraph mentions two widely scattered attempts towards securing the conjecture’s as-yet-missing proof. As a whole, the passage discusses the core issue of Goldbach conjecture remaining “neither proven nor disproven”. The answer choice that does justice to this idea is the correct answer choice.

A. The first paragraph states that “an interesting example of a theorem of the theory of numbers that has been neither proven nor disproven is the so-called Goldbach conjecture”; although it can be inferred that Goldbach conjecture is a theorem of the theory of numbers, the passage does not imply that it is a “fundamental theorem”; besides, because the theorem is not yet proven, it cannot be considered a “fundamental theorem”; even if an inference that Goldbach conjecture is a fundamental theorem of the theory of numbers could be drawn,   it is not the main idea of the passage  . As a whole, the passage discusses the core issue of Goldbach conjecture remaining “neither proven nor disproven”. Incorrect.

B. Trap. The first and second paragraphs discuss Goldbach conjecture as a theorem and   no information about Goldbach as a mathematical genius is provided in the passage  ; even if the passage mentioned that Goldbach was indeed a mathematical genius unparalleled since his time, to mention this idea is not the main idea in the passage. As a whole, the passage discusses the core issue of Goldbach conjecture remaining “neither proven nor disproven”. Incorrect.

C. Trap. The first paragraph mentions that one can easily find that Goldbach conjecture is true as applied to “some simple examples”; it can be inferred that Goldbach conjecture can be verified with simple examples; however, to make this mention is not the main idea in the passage. As a whole, the passage discusses the core issue of Goldbach conjecture remaining “neither proven nor disproven”. Incorrect.

D.  Correct.  As the mind-map suggests, the first paragraph indicates that Goldbach conjecture is neither proven nor disproven and the second paragraph mentions two widely scattered attempts towards securing the conjecture’s as-yet-missing proof; as a whole, the passage discusses the core issue of Goldbach conjecture remaining “neither proven nor disproven”; in other words, Goldbach conjecture is wanting a mathematical verification, as the answer choice mentions.

E. The second paragraph mentions that “the last two steps from Vinogradoff’s four to Goldbach’s two primes seem to be the toughest of all and nobody can tell whether another few years or another few centuries will be required   to prove or disprove   this difficult proposition”; it can be inferred that the proof of Goldbach conjecture is far away; it is incorrect to state that a significant portion of Goldbach conjecture has been proven. Furthermore, as a whole, the passage discusses the core issue of Goldbach conjecture remaining “neither proven nor disproven”. Incorrect.

D is the best choice.   
= = = = = = = = = =

2. Which of the following can be most directly inferred from the usage of the term "last two steps" (see highlighted text)?

A. The author conjectures that the yet-unproven part of Goldbach conjecture can be proven in two steps.
B. It is a term figuratively used to suggest that the seemingly small number of remaining steps are likely the toughest.
C. The proof of "sum of six primes" was two steps before the proof of "sum of four primes".
D. There were three hundred thousand steps in the proof of Goldbach conjecture only two of which are yet to be completed.
E. The number of steps pending in a proof is inversely proportional to the difficulty of those steps.

Highlight Required:   last two steps

Question Type: Inference

Explanation:

The second paragraph makes the mention of the term “last two steps” while discussing two widely scattered attempts taken in the direction of the proof of Goldbach conjecture. The passage states that Schnirelman’s idea of “sum of three hundred thousand primes” was a step closer to the proof of Goldbach conjecture, and that Vinogradoff’s idea of “sum of four primes” is even closer to the proof of Goldbach conjecture. However,   the author doesn’t imply any mathematical relationship between the terms “three hundred thousand primes”, “four primes”, and “two primes”  ; in other words, the term “last two steps” has no mathematical meaning owing to the word “two” in the phrase. The author uses the term “last two steps” to indicate that mathematicians have made “some” progress, but they are far from proving Goldbach conjecture; the author intends to imply a sense of irony by   stating that there are only “last two steps”, but they are the toughest of all   and “nobody can tell whether another few years or another few centuries will be required to prove or disprove this difficult proposition”. The answer choice that does justice to this idea is the correct answer choice.

A. Trap. The second paragraph states that Schnirelman’s idea of “sum of three hundred thousand primes” was a step closer to the proof of Goldbach conjecture, and that Vinogradoff’s idea of “sum of four primes” is even closer to the proof of Goldbach conjecture; however,   the author doesn’t imply any mathematical relationship between the terms “three hundred thousand primes”, “four primes”, and “two primes”  ; in other words, the term “last two steps” has no mathematical meaning owing to the word “two” in the phrase; so, it is incorrect to infer that the author conjectures that the yet-unproven part of Goldbach conjecture can be proven in two steps. Furthermore, the author uses the term “last two steps” to imply a sense of irony by stating that there are only “last two steps”, but they are the toughest of all and “nobody can tell whether another few years or another few centuries will be required to prove or disprove this difficult proposition”. Incorrect.

B.  Correct.  The second paragraph states that Schnirelman’s idea of “sum of three hundred thousand primes” was a step closer to the proof of Goldbach conjecture, and that Vinogradoff’s idea of “sum of four primes” is even closer to the proof of Goldbach conjecture; however,   the author doesn’t imply any mathematical relationship between the terms “three hundred thousand primes”, “four primes”, and “two primes”  ; in other words, the term “last two steps” has no mathematical meaning owing to the word “two” in the phrase; the author uses the term “last two steps” to indicate that mathematicians have made “some” progress, but they are far from proving Goldbach conjecture;   the author intends to imply a sense of irony by stating that there are only “last two steps”, but they are the toughest of all   and “nobody can tell whether another few years or another few centuries will be required to prove or disprove this difficult proposition”; in other words, “last two steps” is a term figuratively used to suggest that the seemingly small number of remaining steps are likely the toughest, as the answer choice mentions.

C. Amateur answer choice. The second paragraph states that Schnirelman’s idea of “sum of three hundred thousand primes” was a step closer to the proof of Goldbach conjecture, and that Vinogradoff’s idea of “sum of four primes” is even closer to the proof of Goldbach conjecture; however,   the author doesn’t imply any mathematical relationship between the terms “three hundred thousand primes”, “four primes”, and “two primes”  ; in other words, the term “last two steps” has no mathematical meaning owing to the word “two” in the phrase; so, it is incorrect to infer that the proof of “sum of six primes” was two steps before the proof of “sum of four primes”. Furthermore, the author uses the term “last two steps” to imply a sense of irony by stating that there are only “last two steps”, but they are the toughest of all and “nobody can tell whether another few years or another few centuries will be required to prove or disprove this difficult proposition”. Incorrect.

D. Amateur answer choice. The second paragraph states that Schnirelman’s idea of “sum of three hundred thousand primes” was a step closer to the proof of Goldbach conjecture, and that Vinogradoff’s idea of “sum of four primes” is even closer to the proof of Goldbach conjecture; however,   the author doesn’t imply any mathematical relationship between the terms “three hundred thousand primes”, “four primes”, and “two primes”  ; in other words, the term “last two steps” has no mathematical meaning owing to the word “two” in the phrase; so, it is incorrect to infer that there were three hundred thousand steps in the proof of Goldbach conjecture only two of which are yet to be completed. Furthermore, the author uses the term “last two steps” to imply a sense of irony by stating that there are only “last two steps”, but they are the toughest of all and “nobody can tell whether another few years or another few centuries will be required to prove or disprove this difficult proposition”. Incorrect.

E. The passage makes   no suggestion regarding any connection between the number of steps pending in a proof and the difficulty of those steps  ; so, it is incorrect to infer that the number of steps pending in a proof are inversely proportional to the difficulty of those steps. Furthermore, the author uses the term “last two steps” to imply a sense of irony by stating that there are only “last two steps”, but they are the toughest of all and “nobody can tell whether another few years or another few centuries will be required to prove or disprove this difficult proposition”. Incorrect.

B is the best choice.   
= = = = = = = = = =

3. Which of the following is most consistent with the world of mathematical theorems as indicated by the passage?

A. All mathematical theorems apart from Goldbach conjecture are proven.
B. Mathematical proofs that can be verified with simple examples are difficult to prove.
C. Even if a theorem can be easily verified, the theorem is expected to be proven.
D. New mathematical explanations cannot offer proof to mathematical theorems previously verified to be true.
E. There is significant uncertainty regarding the proof of all difficult mathematical propositions.

Question Type: Inference

Explanation:

As the mind-map suggests, the first paragraph indicates that Goldbach conjecture is an example of mathematical theorems that are “neither proven nor disproven” and the second paragraph mentions two widely scattered attempts towards securing the conjecture’s as-yet-missing proof. The answer choice that aligns well with the information presented in the passage regarding mathematical theorems is the correct answer choice.

A. The first paragraph mentions that “  An interesting example of a theorem of the theory of numbers that has been neither proven nor disproven   is the so-called Goldbach conjecture”; it can be inferred that Goldbach conjecture is one of the mathematical theorems that are “neither proven nor disproven” and that there are other theorems that lack proof; so, the statement that all mathematical theorems apart from Goldbach conjecture are proven is not consistent with the information presented in the passage. Incorrect.

B. Trap. The first paragraph mentions that one can easily find that Goldbach conjecture is true “as applied to some simple examples”; the second paragraph concludes by stating that “nobody can tell whether another few years or another few centuries will be required to prove or disprove this difficult proposition”; it can be inferred from these references that Goldbach conjecture is easy to verify but extremely difficult to prove or disprove; however,   the passage discusses only Goldbach conjecture and it cannot be inferred that all mathematical proofs that can be verified with simple examples are difficult to prove  , as the answer choice states; because the statement in the answer choice is not consistent with the information presented in the passage, the answer choice is incorrect.

C.  Correct.  The first paragraph mentions that Goldbach conjecture was “proposed in the year 1742” and “One can easily find that it is true as applied to some simple examples”; the paragraph also states that the conjecture is “neither proven nor disproven”; the second paragraph mentions that “In 1931”, Schnirelman came up with "the first constructive step towards securing the desired proof" and “more recently" Vinogradoff's “the sum of four primes” brought us closer to the proof; the second paragraph also mentions that “nobody can tell whether another few years or another few centuries will be required to prove or disprove this difficult proposition”; it can be inferred from these references that   although Goldbach conjecture has been verified to be true, new mathematical explanations have constantly been sought   to prove or disprove the conjecture; in other words, even if a theorem can be easily verified, the theorem is expected to be proven, as the answer choice states; because the statement in the answer choice is consistent with the information presented in the passage, the answer choice is correct.

D. The first paragraph mentions that Goldbach conjecture was “proposed in the year 1742” and “One can easily find that it is true as applied to some simple examples”; the paragraph also states that the conjecture is “neither proven nor disproven”; the second paragraph mentions that “In 1931”, Schnirelman came up with "the first constructive step towards securing the desired proof" and “more recently" Vinogradoff's “the sum of four primes” brought us closer to the proof; the second paragraph also mentions that “nobody can tell whether another few years or another few centuries will be required to prove or disprove this difficult proposition”; it can be inferred from these references that   although Goldbach conjecture has been verified to be true, new mathematical explanations have constantly been sought   to prove or disprove the conjecture; so, the statement that new mathematical explanations cannot offer proofs to mathematical theorems previously verified to be true is not consistent with the information presented in the passage. Incorrect.

E. The second paragraph concludes by stating that “the last two steps from Vinogradoff’s four to Goldbach’s two primes seem to be the toughest of all and nobody can tell whether another few years or another few centuries will be required to prove or disprove this difficult proposition”; it can be inferred that there is uncertainty regarding when Goldbach conjecture will be proven; however, the passage discusses only Goldbach conjecture and   it cannot be inferred that there is significant uncertainty regarding the proof of all difficult mathematical propositions  , as the answer choice states; because the statement in the answer choice is not consistent with the information presented in the passage, the answer choice is incorrect.

C is the best choice.   
= = = = = = = = = =

An interesting example of a theorem of the theory of numbers that has

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An interesting example of a theorem of the theory of numbers that has

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