TITLE 18 § 1301, Mailing lottery tickets or related matter, states as follows,
“Whoever knowingly deposits in the mail, or sends or delivers by mail:
Any letter, package, postal card, or circular concerning any lottery, gift enterprise, or similar scheme offering prizes dependent in whole or in part upon lot or chance;
Any lottery ticket or part thereof, or paper, certificate, or instrument purporting to be or to represent a ticket, chance, share, or interest in or dependent upon the event of a lottery, gift enterprise, or similar scheme offering prizes dependent in whole or in part upon lot or chance;
Any check, draft, bill, money, postal note, or money order, for the purchase of any ticket or part thereof, or of any share or chance in any such lottery, gift enterprise, or scheme;
Any newspaper, circular, pamphlet, or publication of any kind containing any advertisement of any lottery, gift enterprise, or scheme of any kind offering prizes dependent in whole or in part upon lot or chance, or containing any list of the prizes drawn or awarded by means of any such lottery, gift enterprise, or scheme, whether said list contains any part or all of such prizes;
Any article described in section 1953 of this title: Shall be fined under this title or imprisoned not more than two years, or both; and for any subsequent offense shall be imprisoned not more than five years.”
Applying the above statute to your proposed endeavor, it does not appear that your letters will be a violation of the statute.
TITLE 18 § 1431 does not exist.
TITLE 18 § 1341, Frauds and swindles, states as follows,
“Whoever, having devised or intending to devise any scheme or artifice to defraud, or for obtaining money or property by means of false or fraudulent pretenses, representations, or promises, or to sell, dispose of, loan, exchange, alter, give away, distribute, supply, or furnish or procure for unlawful use any counterfeit or spurious coin, obligation, security, or other article, or anything represented to be or intimated or held out to be such counterfeit or spurious article, for the purpose of executing such scheme or artifice or attempting so to do, places in any post office or authorized depository for mail matter, any matter or thing whatever to be sent or delivered by the Postal Service, or deposits or causes to be deposited any matter or thing whatever to be sent or delivered by any private or commercial interstate carrier, or takes or receives therefrom, any such matter or thing, or knowingly causes to be delivered by mail or such carrier according to the direction thereon, or at the place at which it is directed to be delivered by the person to whom it is addressed, any such matter or thing, shall be fined under this title or imprisoned not more than 20 years, or both. If the violation affects a financial institution, such person shall be fined not more than $1,000,000 or imprisoned not more than 30 years, or both.”
It does not appear that your proposed endeavor would violate this statute.