bodyType

customContents

entityId

printArticlePageQuery

filter

article

body

urlAlias

articleSectionsContent

articleSeoArticle

articleSourceHeaderArticle

contentSchemaRenderContent

contentSubHeadlineContent

contentHeadlineContent

mainImagePrintArticleArticle

authorBlockPrintArticleArticle

contentSponsorTagContent

articlePianoPaywallContent

scmpYoutubeContent

hooksTrackPageViewArticle

query printArticlePageQuery(
  $entityId: String!
  $customContents: [CustomContentInput]
  $bodyType: BodyFormatTypeEnum
) {
  article(filter: {entityId: $entityId}) {
    entityId
    body(customContents: $customContents, type: $bodyType) {
      json
    }
    urlAlias
    ...articleSectionsContent
    ...articleSeoArticle_3o8td4
    ...articleSourceHeaderArticle
    ...contentSchemaRenderContent
    ...contentSubHeadlineContent
    ...contentHeadlineContent
    ...mainImagePrintArticleArticle
    ...authorBlockPrintArticleArticle
    ...contentSponsorTagContent
    ...articlePianoPaywallContent
    ...scmpYoutubeContent
    ...hooksTrackPageViewArticle
    id
  }
}

fragment articlePianoPaywallContent on Content {
  __isContent: __typename
  ...hooksPaywallPromiseContent
}

fragment articleSectionsContent on Content {
  __isContent: __typename
  sections {
    value {
      ...helpersCheckIfNewsSectionInFirstIndexSection
      ...entityLink
      name
      id
    }
  }
}

fragment articleSeoArticle_3o8td4 on Article {
  ...hooksArticleSeoArticle
  ...helpersCheckIsPostiesArticle
  ...helpersGetArticleTypeArticle
  entityId
  publishedDate
  updatedDate
  summary {
    text
  }
  subHeadline {
    text
  }
  body(customContents: $customContents, type: $bodyType) {
    text
  }
  keywords
  sections {
    value {
      name
      id
    }
  }
  topics {
    name
    entityId
    id
  }
  urlAlias
  authors {
    name
    id
  }
  images {
    style(filter: {style: "768x768"}) {
      url
    }
  }
  readingTime
}

fragment articleSourceHeaderArticle on Article {
  urlAlias
  ...entityLink
}

fragment authorBlockPrintArticleArticle on Article {
  ...defaultArticleDateContent
  ...authorNamesArticle
}

fragment authorNameAuthor on Author {
  ...entityLink
  name
  entityUuid
}

fragment authorNamesArticle on Article {
  authorLocations
  authors {
    entityId
    location
    ...authorNameAuthor
    id
  }
}

fragment blockquoteQuoteContent on Content {
  __isContent: __typename
  ...contentShareWidgetContent
}

fragment checkersCheckContentContainsStarlingImageContent on Content {
  __isContent: __typename
  topics {
    entityId
    id
  }
}

fragment contentHeadlineContent on Content {
  __isContent: __typename
  headline
  socialHeadline
  ...contentHeadlineTagContent
  ...helpersUseContentSponsorContent
}

fragment contentHeadlineTagContent on Content {
  __isContent: __typename
  ...hooksContentHeadlineTagContent
}

fragment contentNewsletterWidgetContent on Content {
  __isContent: __typename
  topics {
    relatedNewsletters {
      ...helpersMassageToNewsletterItemNewsletter
      id
    }
    id
  }
  sections {
    value {
      relatedNewsletters {
        ...helpersMassageToNewsletterItemNewsletter
        id
      }
      id
    }
  }
  ... on Article {
    relatedNewsletters {
      ...helpersMassageToNewsletterItemNewsletter
      id
    }
  }
}

fragment contentSchemaRenderContent on Content {
  __isContent: __typename
  ...blockquoteQuoteContent
  ...imageContent
  ...inlineWidgetContent
  ...photoGalleryWidgetContent
  ...scmpYoutubeContent
}

fragment contentShareWidgetContent on Content {
  __isContent: __typename
  entityId
  urlAlias
  shortUrl
  socialHeadline
  headline
}

fragment contentSponsorTagContent on Content {
  __isContent: __typename
  ...helpersUseContentSponsorContent
}

fragment contentSubHeadlineContent on Content {
  __isContent: __typename
  subHeadline {
    json
  }
  summary {
    json
  }
  ...contentSchemaRenderContent
}

fragment defaultArticleDateContent on Content {
  __isContent: __typename
  ...hooksArticleDateUtilsContent
}

fragment entityLink on BaseWithApplicationAndUrlAlias {
  __isBaseWithApplicationAndUrlAlias: __typename
  __typename
  entityId
  entityUuid
  urlAlias
  urlRedirect {
    toUrl
    id
  }
  application {
    entityId
    id
  }
  ... on Section {
    fullSectionPath {
      entityId
      id
    }
    parentSection
  }
  ...helpersAppBarVariantBase
  ...helpersApplicationBase
  ...helpersCheckIsPlusArticle
  ...helpersStyleSectionContent
}

fragment helpersAppBarVariantBase on BaseWithApplicationAndUrlAlias {
  __isBaseWithApplicationAndUrlAlias: __typename
  ...helpersCheckIsLiveArticle
  ...helpersCheckIsPlusPage
  ...helpersCheckIsPostiesArticle
  ...helpersCheckIsPostiesPage
  ...helpersCheckIsPostMagazineArticle
  ...helpersCheckIsStyleArticle
  ...helpersSectionContent
  ...helpersStyleSectionContent
}

fragment helpersApplicationBase on BaseWithApplicationAndUrlAlias {
  __isBaseWithApplicationAndUrlAlias: __typename
  ...helpersCheckIsPostiesArticle
  ...helpersCheckIsPostiesPage
  ...helpersCheckIsPlusPage
}

fragment helpersCheckIfNewsSectionInFirstIndexSection on Section {
  name
}

fragment helpersCheckIsAmpEnabledArticle on Article {
  entityId
  types {
    value {
      entityId
      id
    }
  }
  sections {
    value {
      entityId
      id
    }
  }
  disableOffPlatformContent
  sponsorType
  published
  ...helpersCheckIsPlusArticle
  ...helpersCheckIsPostiesArticle
}

fragment helpersCheckIsLiveArticle on Article {
  flattenTypeIds
  liveArticle {
    status
    id
  }
}

fragment helpersCheckIsMainTopicContent on Content {
  __isContent: __typename
  topics {
    entityId
    id
  }
}

fragment helpersCheckIsPlusArticle on Content {
  __isContent: __typename
  ... on Article {
    paywallTypes {
      entityUuid
      id
    }
  }
}

fragment helpersCheckIsPlusPage on Page {
  urlAlias
}

fragment helpersCheckIsPostMagazineArticle on Article {
  sections {
    value {
      entityId
      id
    }
  }
}

fragment helpersCheckIsPostiesArticle on Article {
  sections {
    value {
      entityId
      id
    }
  }
}

fragment helpersCheckIsPostiesPage on Page {
  urlAlias
}

fragment helpersCheckIsStyleArticle on Content {
  __isContent: __typename
  sections {
    value {
      entityId
      id
    }
  }
}

fragment helpersGetArticleTypeArticle on Content {
  __isContent: __typename
  ... on Article {
    types {
      value {
        entityId
        id
      }
    }
  }
}

fragment helpersMassageToNewsletterItemNewsletter on Newsletter {
  entityId
  name
  alternativeName
  description {
    text
  }
  summary {
    text
  }
  homepageDescription {
    text
  }
}

fragment helpersSanitizeArticleTypeEntityIdsArticle on Article {
  types {
    value {
      entityId
      id
    }
  }
}

fragment helpersSanitizeAuthorTypesArticle on Article {
  authors {
    types
    id
  }
}

fragment helpersSanitizedAuthorEntityIdsArticle on Article {
  authors {
    entityId
    id
  }
}

fragment helpersSectionContent on Section {
  fullSectionPath {
    entityId
    id
  }
}

fragment helpersStyleSectionContent on Section {
  fullSectionPath {
    entityId
    id
  }
}

fragment helpersUseContentSponsorContent on Content {
  __isContent: __typename
  topics {
    entityId
    sponsor {
      ...sponsorTagSponsor
      name
      type
      id
    }
    id
  }
  ... on Article {
    sponsorType
  }
  ...helpersCheckIsMainTopicContent
}

fragment hooksArticleDateUtilsContent on Content {
  __isContent: __typename
  createdDate
  publishedDate
  updatedDate
}

fragment hooksArticleSeoArticle on Article {
  ...helpersCheckIsPostiesArticle
  ...helpersCheckIsPostMagazineArticle
  ...helpersCheckIsStyleArticle
  ...hooksContentHeadlineTagContent
  ...helpersCheckIsAmpEnabledArticle
  entityId
  headline
  socialHeadline
  types {
    value {
      entityId
      id
    }
  }
  authors {
    entityId
    id
  }
  images {
    generic_og: style(filter: {style: "og_image_scmp_generic"}) {
      url
    }
    styles(filter: {styles: ["og_image_posties_article", "og_postmag", "og_image_style", "og_image_scmp_analysis", "og_image_scmp_explainer", "og_image_scmp_editorial", "og_image_scmp_fact_check", "og_image_scmp_debate", "og_image_scmp_live", "og_image_scmp_obituary", "og_image_scmp_opinion", "og_image_scmp_review", "og_image_scmp_series", "og_image_scmp_generic"]}) {
      url
      style
    }
    type
  }
  urlAlias
}

fragment hooksContentHeadlineTagContent on Content {
  __isContent: __typename
  __typename
  ... on Article {
    types {
      value {
        name
        entityId
        description {
          text
        }
        id
      }
    }
    ...helpersCheckIsStyleArticle
  }
  flag
  sections {
    value {
      name
      id
    }
  }
  ...hooksContentHelperContent
}

fragment hooksContentHelperContent on Content {
  __isContent: __typename
  ...helpersSanitizeArticleTypeEntityIdsArticle
  ...helpersSanitizeAuthorTypesArticle
  ...helpersSanitizedAuthorEntityIdsArticle
  flag
}

fragment hooksPaywallPromiseContent on Content {
  __isContent: __typename
  entityId
  ...helpersCheckIsPostiesArticle
}

fragment hooksTrackPageViewArticle on Article {
  entityId
  publishedDate
  headline
  sections {
    value {
      name
      id
    }
  }
  urlAlias
  authors {
    name
    id
  }
  entityUuid
}

fragment imageContent on Content {
  __isContent: __typename
  headline
  ...checkersCheckContentContainsStarlingImageContent
}

fragment inlineWidgetContent on Content {
  __isContent: __typename
  ...contentNewsletterWidgetContent
}

fragment mainImagePrintArticleArticle on Article {
  images {
    title
    url
    isSlideshow
    size_652x446: style(filter: {style: "652x446"}) {
      url
    }
  }
}

fragment photoGalleryDialogContent on Content {
  __isContent: __typename
  __typename
  ...contentShareWidgetContent
}

fragment photoGalleryWidgetContent on Content {
  __isContent: __typename
  ...photoGalleryDialogContent
}

fragment scmpYoutubeContent on Content {
  __isContent: __typename
  entityId
  urlAlias
  publishedDate
}

fragment sponsorTagSponsor on Sponsor {
  name
  type
  images {
    url
  }
}


sections

types

disableOffPlatformContent

sponsorType

published

headline

socialHeadline

authors

images

publishedDate

updatedDate

summary

subHeadline

keywords

topics

readingTime

authorLocations

entityUuid

NEW YORK TIMES

Three and five are prime numbers - that is, they are divisible only by one and by themselves. So are five and seven.
And 11 and 13. And for each of these pairs of prime numbers, the difference is two.
Mathematicians have long believed that there are an infinite number of such pairs, called twin primes, meaning that there will always be a larger pair than the largest one found. This supposition, the so-called Twin Prime Conjecture, is not necessarily obvious. As numbers get larger, prime numbers become sparser among vast expanses of divisible numbers. Yet still - occasionally, rarely - two consecutive odd numbers will both be prime, the conjecture asserts.
The proof has been elusive.
But last month, a paper from a little-known mathematician arrived "out of the blue" at the journal Annals of Mathematics, said Peter Sarnak, a professor of mathematics at Princeton University and the Institute for Advanced Study and a former editor at the journal, which plans to publish it.
The paper, by Zhang Yitang of the University of New Hampshire, does not prove that there are an infinite number of twin primes, but it does show an infinite number of prime pairs whose separation is less than a finite upper limit - 70 million, for now. (Zhang used 70 million in his proof - basically an arbitrary large number where his equations work.)
"It's a deep insight," Sarnak said. "It's a deep result."
Zhang said he had been working on the Twin Prime Conjecture for years and, like everyone else, failed.
"I tried everything," said Zhang, who was originally from China. Then, last July, "just very suddenly, an idea came to my mind", Zhang said. "I was confident in this way I could prove it."
It took him another six months to fill in the details, but he appears to be right. The paper has been accepted pending some small revisions. "It's remarkable the speed this paper was dealt with," Sarnak said.
Zhang's proof takes advantage of a 2005 paper by Daniel Goldston of San Jose State University, Janos Pintz of the Alfred Renyi Institute of Mathematics in Budapest and Cem Yildirim of Bogazici University in Istanbul, which had shown there would always be pairs of primes closer than the average distance between two primes.
Still, in mathematics, closer does not necessarily mean two numbers away, and experts were unable to make further progress on the conjecture. "People tried, and after a few years, it seemed this was really far away," Sarnak said.
Zhang also used techniques developed in the 1980s by Henryk Iwaniec of Rutgers, Enrico Bombieri of the Institute for Advanced Study and John B. Friedlander of the University of Toronto, adding his own ingenuity to tie everything together in a way others had been unable to.
"He got it," said Iwaniec, who has read Zhang's paper. "There's no question about it."
The next step is reducing that 70 million separation, and Zhang said "that should be very simple".
But experts said bringing it all the way down to two - the full Twin Prime Conjecture - would probably require more mathematical breakthroughs.


Mathematician makes prime number breakthrough

Zhang Yitang

Lifestyle

Tech

'Deep insight' takes mathematicians nearer goal of proving Twin Prime Conjecture


Three and five are prime numbers - that is, they are divisible only by one and by themselves. So are five and seven. And 11 and 13. And for each of these pairs of prime numbers, the difference is two. Mathematicians have long believed that there are an infinite number of such pairs, called twin primes, meaning that there will always be a larger pair than the largest one found.
