# sufficient condition of identical congruence

Theorem. Let $f(X):={a}_{n}{X}^{n}+\mathrm{\dots}+{a}_{1}X+{a}_{0}$ be a polynomial^{} in $X$ with integer coefficients ${a}_{i}$ and $m$ a positive integer. If the congruence^{}

$f(x)\equiv \mathrm{\hspace{0.33em}0}\phantom{\rule{veryverythickmathspace}{0ex}}(modm)$ | (1) |

is satisfied by $m$ successive integers $x$, then it is satisfied by all integers $x$, in other words it is an identical congruence.

*Proof.* There is an integer ${x}_{0}$ such that (1) is satisfied by

$$x:={x}_{0}+1,{x}_{0}+2,\mathrm{\dots},{x}_{0}+m.$$ |

But these values form a complete residue system^{} modulo $m$. Thus, if $x$ is an arbitrary integer, one has

$$x\equiv {x}_{0}+r\phantom{\rule{veryverythickmathspace}{0ex}}(modm)\mathit{\hspace{1em}}\text{where}\mathrm{\hspace{0.33em}\hspace{0.33em}1}\leqq r\leqq m.$$ |

This implies

$${a}_{i}{x}^{i}\equiv {a}_{i}{({x}_{0}+r)}^{i}\phantom{\rule{veryverythickmathspace}{0ex}}(modm)\mathit{\hspace{1em}}\text{for}i=0,\mathrm{\hspace{0.17em}1},\mathrm{\dots},n$$ |

and consequently

$$\underset{f(x)}{\underset{\u23df}{\sum _{i=0}^{n}{a}_{i}{x}^{i}}}\equiv \sum _{i=0}^{n}{a}_{i}{({x}_{0}+r)}^{i}=f({x}_{0}+r)\equiv \mathrm{\hspace{0.33em}0}\phantom{\rule{veryverythickmathspace}{0ex}}(modm).$$ |

Accordingly, (1) is true for any integer $x$, Q.E.D.

Note. Though the congruence (1) is identical, it need not be a question of a formal congruence

$f(X)\underset{\xaf}{\equiv}\mathrm{\hspace{0.33em}0}\phantom{\rule{veryverythickmathspace}{0ex}}(modm),$ | (2) |

i.e. all coefficients ${a}_{i}$ need not be congruent to 0 modulo $m$.

Title | sufficient condition of identical congruence |
---|---|

Canonical name | SufficientConditionOfIdenticalCongruence |

Date of creation | 2013-03-22 18:56:03 |

Last modified on | 2013-03-22 18:56:03 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 11C08 |

Classification | msc 11A07 |

Related topic | Sufficient |

Related topic | CongruenceOfArbitraryDegree |

Related topic | PolynomialCongruence |