Cumulative voting is a multiple-winner voting method intended to promote more proportional representation than winner-take-all elections. Cumulative voting is used in corporate governance, where it is mandated by some U. S. states. It was used to elect the Illinois House of Representatives from 1870 until its repeal in 1980 and used in England in the late 19th century to elect some school boards; as of March 2012, more than fifty communities in the United States use cumulative voting, all resulting from cases brought under the National Voting Rights Act of 1965. Among them are Peoria, Illinois for half of its city council, Chilton County, Alabama for its county council and school board, Amarillo, for its school board and College Board of Regents. Courts sometimes mandate its use as a remedy in lawsuits brought under the Voting Rights Act in the United States. A form of cumulative voting has been used by group facilitators as a method to collectively prioritize options, for example ideas generated from a brainstorming session within a workshop.
This approach is described as “multi-voting” and was derived from the nominal group technique and is one of many tools suggested within the Six Sigma business management strategy. A cumulative voting election permits voters in an election for more than one seat to put more than one vote on a preferred candidate; when voters in the minority concentrate their votes in this way, it increases their chances of obtaining representation in a legislative body. This is different from bloc voting, where a voter may not vote more than once for any candidate, 51% of voters can control 100% of representation. Ballots used for cumulative voting differ both in the ways voters mark their selections and in the degree to which voters are permitted to split their own vote; the simplest ballot uses the equal and cumulative voting method, where a voter marks preferred candidates, as in bloc voting, votes are automatically divided evenly among those preferred candidates. Voters are unable to specify a differing level of support for a more preferred candidate, giving them less flexibility although making it tactically easier to support a slate of candidates.
A more common and more complex cumulative ballot uses a points method. Under this method, voters are given an explicit number of points to distribute amongst candidates on a single ballot; this is done with a voter making a mark for each point beside the desired candidate. A similar method is to have the voter write in the desired number of points next to each candidate; this latter approach is used for corporate elections involving a large number of points on a given ballot, where the voter is given one set of points for each votable share of stock he has in the company. Unless an appropriately programmed electronic voting system is used, this write-in ballot type burdens the voter with ensuring that his point allocations add up to his allotted sum; when used as a facilitation technique for group decision-making this process is called “multi-voting”. Participants are given points which they can apply among a list of options; because dot stickers are used for multi-voting, the process is often called dot voting.
In typical cumulative elections using the points method, the number of points allotted to a voter is equal to the number of winning candidates. This allows a voter to express some support for all winning candidates. With only one point the method becomes equivalent to a single non-transferable vote in a first-past-the-post method. Other than general egalitarian concerns of electoral equality, there is nothing in this method that requires each voter to be given the same number of points. If certain voters are seen as more deserving of influence, for example because they own more shares of stock in the company, they can be directly assigned more points per voter; this explicit method of granting particular voters more influence is advocated for governmental elections outside corporate management because the voters are members of an oppressed group. Unlike choice voting where the numbers represent the order of a voter's ranking of candidates, in cumulative votes the numbers represent quantities.
While giving voters more points may appear to give them a greater ability to graduate their support for individual candidates, it is not obvious that it changes the democratic structure of the method. The most flexible ballot allows a full vote to be divided in any fraction among all candidates, so long as the fractions add to less than or equal to 1. Advocates of cumulative voting argue that political and racial minorities deserve better representation. By concentrating their votes on a small number of candidates of their choice, voters in the minority can win some representation — for example, a like-minded grouping of voters, 20% of a city would be well-positioned to elect one out of five seats. Both forms of cumulative voting achiev
Assassination of Abraham Lincoln
Abraham Lincoln, the 16th President of the United States, was assassinated by well-known stage actor John Wilkes Booth on April 14, 1865, while attending the play Our American Cousin at Ford's Theatre in Washington, D. C. Shot in the head as he watched the play, Lincoln died the following day at 7:22 am, in the Petersen House opposite the theater, he was the first U. S. president to be assassinated, Lincoln's funeral and burial marked an extended period of national mourning. Occurring near the end of the American Civil War, the assassination was part of a larger conspiracy intended by Booth to revive the Confederate cause by eliminating the three most important officials of the United States government. Conspirators Lewis Powell and David Herold were assigned to kill Secretary of State William H. Seward, George Atzerodt was tasked with killing Vice President Andrew Johnson. Beyond Lincoln's death the plot failed: Seward was only wounded and Johnson's would-be attacker lost his nerve. After a dramatic initial escape, Booth was killed at the climax of a 12-day manhunt.
Powell, Herold and Mary Surratt were hanged for their roles in the conspiracy. John Wilkes Booth, born in Maryland into a family of prominent stage actors, had by the time of the assassination become a famous actor and national celebrity in his own right, he was an outspoken Confederate sympathizer. In March 1864, Ulysses S. Grant, commander of the Union armies, suspended the exchange of prisoners of war with the Confederate Army to increase pressure on the manpower-starved South. Booth conceived a plan to kidnap Lincoln in order to blackmail the North into resuming prisoner exchanges,:130–4 and recruited Samuel Arnold, George Atzerodt, David Herold, Michael O'Laughlen, Lewis Powell, John Surratt to help him. Surratt's mother, Mary Surratt, left her tavern in Surrattsville and moved to a house in Washington, D. C. where Booth became a frequent visitor. While Booth and Lincoln were not acquainted, Lincoln had seen Booth at Ford's in 1863.:419 After the assassination, actor Frank Mordaunt wrote that Lincoln admired Booth, whom Lincoln had invited to visit the White House.
Booth attended Lincoln's second inauguration on March 4, writing in his diary afterwards: "What an excellent chance I had, if I wished, to kill the President on Inauguration day!":174,437n.41On March 17, Booth and the other conspirators planned to abduct Lincoln as he returned from a play at Campbell Military Hospital. But Lincoln did not go instead attending a ceremony at the National Hotel. On April 3, Virginia, the Confederate capital, fell to the Union Army. On April 9 the General-in-Chief of the Confederate States Army Robert E. Lee and his Army of Northern Virginia surrendered to the Commanding General of the United States Army Ulysses S. Grant and his Army of the Potomac after the Battle of Appomattox Court House. Confederate President Jefferson Davis and other Confederate officials had fled, but Booth continued to believe in the Confederate cause and sought a way to salvage it.:728 There are various theories about Booth's motivations. In a letter to his mother, he wrote of his desire to avenge the South.
Doris Kearns Goodwin has endorsed the idea that another factor was Booth's rivalry with his well-known older brother, actor Edwin Booth, a loyal Unionist. David S. Reynolds believes Booth admired the abolitionist John Brown. On April 11, Booth attended Lincoln's speech at the White House in which Lincoln promoted voting rights for blacks; that is the last speech he will give." He urged Lewis Powell to shoot Lincoln on the spot, when Powell refused for fear of the crowd, said to David Herold, "By God, I'll put him through.":91 According to Ward Hill Lamon, three days before his death Lincoln related a dream in which he wandered the White House searching for the source of mournful sounds: I kept on until I arrived at the East Room, which I entered. There I met with a sickening surprise. Before me was a catafalque, on which rested a corpse wrapped in funeral vestments. Around it were stationed soldiers who were acting as guards. "Who is dead in the White House?" I demanded of one of the soldiers, "The President," was his answer.
For months Lincoln had looked pale and haggard, but on the morning of the assassination he told people how happy he was. First Lady Mary Todd Lincoln felt such talk could bring bad luck.:346 Lincoln told his cabinet that he had dreamed of being on a "singular and indescribable vessel, moving with great rapidity toward a dark and indefinite shore", that he'd had the same dream before "nearly every great and important event of the War" such as the victories at Antietam, Murfreesboro and Vicksburg. On April 14, Booth's morning started at midnight, he wrote his mother that all was well, but that he was "in haste". In his diary, he wrote that "Our cause being lost, something decisive and great must be done".:728:346While visiting Ford's Theatre around noon to pick up his mail, Booth learned that Lincoln and Grant were to see Our American Cousin there that night. This provided him with an good
Governor of Illinois
The Governor of Illinois is the chief executive of the State of Illinois, the various agencies and departments over which the officer has jurisdiction, as prescribed in the state constitution. It is votes being cast by popular suffrage of residents of the state; the governor is responsible for enacting laws passed by the Illinois General Assembly. Illinois is one of 14 states; the governor is commander-in-chief of the state's land and sea forces, when they are in state service. The current governor is Democrat J. B. Pritzker, who took office on January 14, 2019; the term of office of Governor of Illinois is four years, there is no limit on the number of terms a governor may serve. Inauguration takes place on the second Monday in January following a gubernatorial election. A single term ends four years later. A governor is required to be: at least twenty-five years old a United States citizen a resident of Illinois for three years prior to election If the incumbent governor is no longer able or permitted to fulfill the duties of the office of governor, the line of succession is as follows: The governor is allowed the occupancy of the Illinois Governor's Mansion in Springfield, the state capital.
Its first occupant was Governor Joel Aldrich Matteson, who took residence at the mansion in 1855. It is one of three oldest governor's residences in continuous use in the United States; the governor is given the use of an official residence on the state fair grounds located in Springfield. Governors have traditionally used this residence part of the year. However, some governors, such as Rod Blagojevich, have chosen to not use the governor's homes as their primary residence, instead commuting either by car or plane to Springfield from their home cities. Many Chicago-based governors have done much of their business out of the governor's office in Chicago's James R. Thompson Center, an office building owned by the state named for former governor James R. Thompson Illinois' longest-serving governor. Six Illinois governors have been charged with crimes after their governorships. Len Small, governor from 1921 to 1929, was indicted in office for corruption, he was acquitted. Among his defense lawyers was a former governor, Joseph W. Fifer, who asserted in pre-trial hearings, that the governorship has the divine right of kings.
William G. Stratton, governor from 1953 to 1961, was acquitted of tax evasion in 1965. Otto Kerner, Jr. governor from 1961 to 1968. He was prosecuted by future Illinois governor Jim Thompson. Daniel Walker, governor from 1973 to 1977, was involved in the savings and loan scandals and convicted of federal crimes related to fraudulent loans to himself from his own First American Savings & Loan Association of Oak Brook, he was sentenced to seven years in prison with five years of probation following his release. George Ryan, governor from 1999 to 2003, was convicted in 2006 of corruption related to his time as Illinois Secretary of State in the 1990s, when commercial driver's licenses were issued to unqualified truckers in exchange for bribes, one of the truckers was involved in a crash that killed six children. Former governor Jim Thompson, whom Ryan had served under as Lieutenant Governor of Illinois in the 1980s, was manager of the law firm that defended Ryan. Ryan was released in 2013. Rod Blagojevich, governor from 2003 to 2009, Ryan's successor, was impeached and removed from office by the Illinois General Assembly in a unanimous vote in January 2009 after being tied to multiple "pay to play" schemes, including attempting to sell the former Senate seat of then-President-elect Barack Obama.
In August 2010, he was convicted of lying to the FBI in connection with the investigation, but the jury deadlocked on 23 other charges. Blagojevich was retried on 20 counts from his 2010 trial and on June 27, 2011, Blagojevich was convicted on 17 counts of fraud, acquitted on one count and the jury was hung on two. On December 7, 2011, Blagojevich was sentenced to 14 years in prison. List of Governors of Illinois 1.α Current governor of Illinois Bruce Rauner, independently wealthy, has stated that he would only accept $1 in salary. In 2015, the Council of State Governments reported that Rauner had returned all but $1 of his salary to the State of Illinois. However, the pay rate for the title of governor in Illinois remains at $177,412. Illinois Office of the Governor Illinois Executive Mansion Burial places of Illinois Governors Article V in the Illinois Constitution list of government help in Illinois
John J. Cullerton is an American politician, a Democratic member of the Illinois Senate, representing the 6th district since his appointment in 1991, he was elected President of the Illinois Senate in 2009. Cullerton is involved in an ongoing corruption scandal in which he is accused of pressuring the Cook County Department of Transportation and Highways to pave public green space at taxpayer expense to enable a real estate development that he co-owns. Cullerton is a native of Chicago, he received his bachelor's degree in political science from Loyola University of Chicago, where he earned his law degree. After graduating from law school, Cullerton served as a Chicago Assistant Public Defender, he went on to work at the law firm of Haber. In 1979, he was elected to the Illinois General Assembly where he served for twelve years as a member of the House of Representatives, he served as Democratic Floor Leader. According to Cullerton's campaign website, he sponsored the most bills and had the most bills passed of all legislators in the 93rd and 94th General Assemblies.
After being appointed to fill Dawn Clark Netsch's seat in 1991, Cullerton was elected to the state senate in 1992 where he was appointed Senate Majority Caucus Whip. Cullerton has been recognized for sponsoring more bills than any other legislator and having more signed into law by the governor. Cullerton was chosen as the senate president by the Senate Democratic Caucus on November 19, 2008 to begin serving in 2009, replacing the retiring Emil Jones, his first legislative priority as senate president was to pass the first Capital Bill in 10 years, which allocated $31 billion for public works projects and created tens of thousands of jobs in Illinois Public Act 096-0036. Cullerton led the senate during the impeachment trial, subsequent removal, of former Governor Rod Blagojevich. Cullerton served as a delegate to the 2012 Democratic National Convention. Senator Cullerton supported SB-1; the Illinois Supreme Court found these legislative changes to be unconstitutional. As the Illinois Supreme Court ruling stated: "These modifications to pension benefits unquestionably diminish the value of the retirement annuities the members…were promised when they joined the pension system.
Accordingly, based on the plain language of the Act, these annuity-reducing provisions contravene the pension protection clause's absolute prohibition against diminishment of pension benefits and exceed the General Assembly's authority," Since the rejection of the constitutionality of SB-1 Senator Cullerton has continued to support the reduction of pension benefits of Illinois State employees. In May 2017, Cullerton led the state Senate Democrats in passing a bill that increased the state individual income tax from 3.75 to 4.95 percent, along with a number of tax increases on businesses. The tax increases, if signed into law, were projected to bring in $4.453 billion from individuals and another $1 billion from businesses. In May 2017, Cullerton intervened in a land dispute outside of his district when he advocated, on behalf of the Keefe Family Trust, to pave over a section of publicly owned wetland to build a 28 foot long driveway, which would require killing 48 mature trees in a small old-growth forest.
Despite the opposition and objections of the Village of Wilmette, the City of Evanston, the publicly operated Canal Shores Golf Course and numerous community organizations, Cullerton met with local officials on multiple occasions to argue in favor of a driveway to access a landlocked parcel so the Keefe Family Trust could build a subdivision of three houses. The parcel had been landlocked. Cullerton serves part-time as an Illinois state senator. Fagel Haber merged with Thompson Coburn LLP in 2007, Cullerton continues as a partner practicing in the areas of government relations, licensing, real estate tax assessment, nonprofit law. Cullerton and his wife, have five children together: Maggie, John III, Kyle, Josephine. Biography and committees at the 98th Illinois General Assembly By session: 98th, 97th, 96th, 95th, 94th, 93rd Illinois Senate President John Cullerton legislative website Senate President John J. Cullerton at Illinois Senate Democrats Profile at Vote Smart Collected news and commentary at the Chicago Tribune
Democratic Party of Illinois
The Democratic Party of Illinois is the affiliate of the U. S. Democratic Party in the U. S. state of Illinois. It is the oldest extant state party in Illinois and, along Republican Parties, one of just two recognized parties in the state; the Democratic Party of Illinois took shape during the late 1830s. Prior to that time, Illinois did not have organized political parties; as the Democratic and Whig Parties began to form at the national level during the late 1820s and 1830s, Illinois politicians began sorting themselves accordingly and, in the summer of 1837, leading Democrats met to lay the groundwork for a Democratic Party organization in the state. Before 2010, the party had been successful in statewide elections for the past decade. In 1992, Carol Moseley Braun became the first African American woman to be elected to the United States Senate, her election marked the first time Illinois had elected a woman, the first time a black person was elected as a Democratic Party candidate to the United States Senate.
A second African American Democratic Senator, Barack Obama was elected in 2004, elected President of the United States in 2008. The Democrats hold supermajorities in both the Illinois Senate and Illinois House of Representatives; the Democratic Party of Illinois is run by a Democratic State Central Committee of 38 members, two from each of the state's 19 congressional districts. The Central Committee has four officers: a chairman, a vice-chair, a secretary, a treasurer. Calvin Sutker of Skokie served as state party chairman until 1986 when he lost his committeeman seat to reform Democrat Jeffrey Paul Smith. Sutker was succeeded by Vince Demuzio, who served from 1986 to 1990 and is credited with rebuilding the Illinois Democratic Party. Demuzio was defeated by the then-chief of staff for Illinois House Speaker Michael Madigan, Gary LaPaille. Madigan himself is the current Chairman of the Democratic Party of Illinois; the Cook County Democratic Party represents voters in 50 wards in the city of Chicago and 30 suburban townships of Cook County.
The organization has dominated Chicago politics since the 1930s. It relies on a tight organizational structure of township committeemen to elect candidates. Illinois Republican Party Political party strength in Illinois Democratic Party of Illinois
Barack Hussein Obama II is an American attorney and politician who served as the 44th president of the United States from 2009 to 2017. A member of the Democratic Party, he was the first African American, he served as a U. S. senator from Illinois from 2005 to 2008. Obama was born in Hawaii. After graduating from Columbia University in 1983, he worked as a community organizer in Chicago. In 1988, he enrolled in Harvard Law School, where he was the first black president of the Harvard Law Review. After graduating, he became a civil rights attorney and an academic, teaching constitutional law at the University of Chicago Law School from 1992 to 2004, he represented the 13th district for three terms in the Illinois Senate from 1997 until 2004 when he ran for the U. S. Senate, he received national attention in 2004 with his March primary win, his well-received July Democratic National Convention keynote address, his landslide November election to the Senate. In 2008, he was nominated for president a year after his campaign began and after a close primary campaign against Hillary Clinton.
He was elected over Republican John McCain and was inaugurated on January 20, 2009. Nine months he was named the 2009 Nobel Peace Prize laureate. Regarded as a centrist New Democrat, Obama signed many landmark bills into law during his first two years in office; the main reforms that were passed include the Patient Protection and Affordable Care Act, the Dodd–Frank Wall Street Reform and Consumer Protection Act, the Don't Ask, Don't Tell Repeal Act of 2010. The American Recovery and Reinvestment Act of 2009 and Tax Relief, Unemployment Insurance Reauthorization, Job Creation Act of 2010 served as economic stimulus amidst the Great Recession. After a lengthy debate over the national debt limit, he signed the Budget Control and the American Taxpayer Relief Acts. In foreign policy, he increased U. S. troop levels in Afghanistan, reduced nuclear weapons with the United States–Russia New START treaty, ended military involvement in the Iraq War. He ordered military involvement in Libya in opposition to Muammar Gaddafi.
He ordered the military operations that resulted in the deaths of Osama bin Laden and suspected Yemeni Al-Qaeda operative Anwar al-Awlaki. After winning re-election by defeating Republican opponent Mitt Romney, Obama was sworn in for a second term in 2013. During this term, he promoted inclusiveness for LGBT Americans, his administration filed briefs that urged the Supreme Court to strike down same-sex marriage bans as unconstitutional. He advocated for gun control in response to the Sandy Hook Elementary School shooting, indicating support for a ban on assault weapons, issued wide-ranging executive actions concerning climate change and immigration. In foreign policy, he ordered military intervention in Iraq in response to gains made by ISIL after the 2011 withdrawal from Iraq, continued the process of ending U. S. combat operations in Afghanistan in 2016, promoted discussions that led to the 2015 Paris Agreement on global climate change, initiated sanctions against Russia following the invasion in Ukraine and again after Russian interference in the 2016 United States elections, brokered a nuclear deal with Iran, normalized U.
S. relations with Cuba. During his term in office, America's reputation in global polling improved. Evaluations of his presidency among historians, political scientists, the general public place him among the upper tier of American presidents. Obama left office and retired in January 2017 and resides in Washington, D. C. A December 2018 Gallup poll found Obama to be the most admired man in America for an unprecedented 11th consecutive year, although Dwight D. Eisenhower was selected most admired in twelve non-consecutive years. Obama was born on August 4, 1961, at Kapiolani Medical Center for Women and Children in Honolulu, Hawaii, he is the only president, born outside of the contiguous 48 states. He was born to a black father, his mother, Ann Dunham, was born in Kansas. His father, Barack Obama Sr. was a Luo Kenyan from Nyang'oma Kogelo. Obama's parents met in 1960 in a Russian language class at the University of Hawaii at Manoa, where his father was a foreign student on a scholarship; the couple married in Hawaii, on February 2, 1961, six months before Obama was born.
In late August 1961, Barack and his mother moved to the University of Washington in Seattle, where they lived for a year. During that time, the elder Obama completed his undergraduate degree in economics in Hawaii, graduating in June 1962, he left to attend graduate school on a scholarship at Harvard University, where he earned an M. A. in economics. Obama's parents divorced in March 1964. Obama Sr. returned to Kenya in 1964, where he married for a third time and worked for the Kenyan government as the Senior Economic Analyst in the Ministry of Finance. He visited his son in Hawaii only once, at Christmas time in 1971, before he was killed in an automobile accident in 1982, when Obama was 21 years old. Recalling his early childhood, Obama said, "That my father looked nothing like the people around me – that he was black as pitch, my mother white as milk – registered in my mind." He described his struggles as a young adult to reconcile social perceptions of his multira
In mathematics, parity is the property of an integer's inclusion in one of two categories: or odd. An integer is if it is divisible by two and odd if it is not even. For example, 6 is because there is no remainder when dividing it by 2. By contrast, 3, 5, 7, 21 leave a remainder of 1 when divided by 2. Examples of numbers include −4, 0, 82 and 178. In particular, zero is an number; some examples of odd numbers are −5, 3, 29, 73. A formal definition of an number is that it is an integer of the form n = 2k, where k is an integer, it is important to realize that the above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings; the sets of and odd numbers can be defined as following: Even = Odd = A number expressed in the decimal numeral system is or odd according to whether its last digit is or odd.
That is, if the last digit is 1, 3, 5, 7, or 9 it is odd. The same idea will work using any base. In particular, a number expressed in the binary numeral system is odd if its last digit is 1 and if its last digit is 0. In an odd base, the number is according to the sum of its digits – it is if and only if the sum of its digits is even; the following laws can be verified using the properties of divisibility. They are a special case of rules in modular arithmetic, are used to check if an equality is to be correct by testing the parity of each side; as with ordinary arithmetic and addition are commutative and associative in modulo 2 arithmetic, multiplication is distributive over addition. However, subtraction in modulo 2 is identical to addition, so subtraction possesses these properties, not true for normal integer arithmetic. Even ± = even; the division of two whole numbers does not result in a whole number. For example, 1 divided by 4 equals 1/4, neither nor odd, since the concepts and odd apply only to integers.
But when the quotient is an integer, it will be if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad, to be neither odd nor even; some of this sentiment survived into the 19th century: Friedrich Wilhelm August Fröbel's 1826 The Education of Man instructs the teacher to drill students with the claim that 1 is neither nor odd, to which Fröbel attaches the philosophical afterthought, It is well to direct the pupil's attention here at once to a great far-reaching law of nature and of thought. It is this, that between two different things or ideas there stands always a third, in a sort of balance, seeming to unite the two. Thus, there is here between odd and numbers one number, neither of the two. In form, the right angle stands between the acute and obtuse angles. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this and other important laws. Integer coordinates of points in Euclidean spaces of two or more dimensions have a parity defined as the parity of the sum of the coordinates.
For instance, the face-centered cubic lattice and its higher-dimensional generalizations, the Dn lattices, consist of all of the integer points whose sum of coordinates is even. This feature manifests itself in chess, where the parity of a square is indicated by its color: bishops are constrained to squares of the same parity; this form of parity was famously used to solve the mutilated chessboard problem: if two opposite corner squares are removed from a chessboard the remaining board cannot be covered by dominoes, because each domino covers one square of each parity and there are two more squares of one parity than of the other. The parity of an ordinal number may be defined to be if the number is a limit ordinal, or a limit ordinal plus a finite number, odd otherwise. Let R be a commutative ring and let I be an ideal of R whose index is 2. Elements of the coset 0 + I may be called while elements of the coset 1 + I may be called odd; as an example, let R = Z be the localization of Z at the prime ideal.
An element of R is or odd if and only if its numerator is so in Z. The numbers form an ideal in the ring of integers, but the odd numbers do not — this is clear from the fact that the identity element for addition, zero, is an element of the numbers only. An integer is if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, odd if it is congruent to 1 modulo 2. All prime numbers are odd, with one exception: the prime number 2. All known perfect numbers are even. Goldbach's conjecture states that every integer greater than 2 can be represented as a sum of two prime numbers. Modern computer calculations have shown this conjecture to